I live in a part of the world where the sun is a major heat source in winter, and the daytime temperature standing in the sun can be as much as 10-15 degrees warmer than what you feel standing in the shade.

The building where I live is stone, which retains heat, so you might not get the same benefit. We also have metal shutters, which keeps cold winter air at a bit of a remove from the windows at night. If you have thermal drapes and pull them over the sliders and other windows once the sun is no longer there, that should also help.

How thick are your rugs?

I have a brick house. With cork bottom vinyl flooring. I recently just bought a thick shag area rug 8x10 and looking to see if I can put that carpet insulation underneath, the room is big enough that I can put the same rug in the other section. Our winters here can be brutal. My backyard is so hot in the summer with the southwest sun exposure. I think maybe that is why the awnings are there from previous owner. I also have blackout shades added as I cant stand vertical blinds.

Yes, awnings do prevent heat gain. Remove it for a year and see how the room does. Also, consider adding a booster fan in your floor vent: Google "floor vent booster fan". Sample: https://www.amazon.com/AC-Infinity-Register-Thermostat-Control/dp/B0792QR5YT/

I would definitely try removing the awning and in the autumn and letting the brick heat up during the day, plus letting the sun into the room. If you can close off the family room to the rest of the house while the sun is beating on it, all the better. More than blackout shades, you may need thermal drapes to trap the heat in the house once the sun is down. if it’s not a tripping hazard, try having an added throw rug under chairs or sofas to catch any drafts. You can sometimes find old pelt rugs or thick wool ones in vintage stores. Synthetic materials are not as warm as natural ones. Skirted sofas and chairs can be better than ones with bare legs up off the floor.

Split levels are notorious when it comes to HVAC issues. The most effective solution depends on your budget. Yes, remove the awning to encourage solar gain and install a thick rug. "Pushing air" into this room is a waste of money, consult with your HVAC Pro as to the feasibility of sealing off the trunk line that supplies this area. Best fix is installing a floating floor, with insulation batts, over the existing floor...then choosing an effective heat source for the area. Keep in mind that hot air rises - being able to "contain" the heat within the area (so it doesn't migrate to the upper level) is also necessary. https://www.youtube.com/watch?v=DR8SqpguwzM

The flooring idea you suggested, we did look into. The demo to remove 2 layers of tile that existed was costly, and to much of an undertaking to do ourselves with an infant. Along with costs to install in an area about 600 square feet. We went the less expensive route and installed a vinyl floor with cork bottom that acts as a floating floor. It really did make a difference on our cold feet, obviously not 100%. We left the wood paneling on the walls and painted over them, and added wainscoting. Doing that did help a bit. Not in the budget unfortunately with removing flooring or even gutting the room to add new insulation. Ugh! I will remove the awning and see what happens :)

Have you looked into window replacements? None of these things are cheap, I realize.

Do you have a cold wind persistently blowing from one direction? Trees can help….

https://myperfectplants.com/2019/05/14/best-windbreak-trees/

Yes! I actually have someone coming out next week to quote windows for the whole house. Still working with home depot to get new exterior doors. I love my house and it has great bones, the updating is causing more expenses as we go along we are finding things needing replaced the right way versus cheap fixes. Who ever saw a toilet grouted in the floor tile in the bathroom? We had to break down that toilet to add a new one, good thing we did that o ring was in desperate need of a change.

IMO use solar shades to keep the heat out in the hot months and roll them up in the rest of theyear to allow as much sun as possible to help heat the space up If you post some pics of your house it might help too.New double or even triple paned windows will for sure help keep jwta ever heat you have inside . Other peoples DIY leftovers are always a PITA.

Just to comment extra, some of the fixes are way to costly like adding a new floor like the video. I already paid $2500 in plumbing repair and replacement that just happened, I have to get an electrician in here to fix some home made electrical stuff. Paying landscapers almost $2k to fix the mess on the side of tge house that is a mixture of concrete blocks/bricks/and pavers, to essential grass off a section and a ton of rocks. I am recently a single mom on top of it.

We also had the problemof a cold slab floor in our former house. It was finally solved when I bought a carpet remnant bound to fit and placed it over a rug pad the same size. This prevented the radiation of the cold into the room. The room was something like 22X 12 and the rug remnant cost less than $300. This seems as if it might be an affordable option for you. However, with remnants you don't have full choice of color, but I found a nice tweed that did well for us.

Awnings or anything else blocking the sunlight does matter.

A tree that shades in summer and sheds its leaves in winter to let the sun in can be a huge help in naturally controlling the temperature of a room .

Whether or not you have single or double pane windows matters as well.

Is your exterior wall/foundation around the downstairs waterproofed? That can matter as well. Damp soil around a basement can have a cooling effect and have potential for problems with mold or mildew.

Budget permitting, consider new flooring for the room. Seal/waterproof the concrete, add a kind of flooring (not carpet) under which you can add an underlayment -- a padding that can help block cold from the ground below the concrete... even if that means you have 7'9" ceilings instead of 8' ceilings. I'd seriously consider the cork underlayment rather than a "carpet" type of underlayment that could hold moisture ... but I'm certainly not an expert .

The room will definatly gain heat if sun is alowed to hit glass door. Much has been discussed about floor covering. To get maximunm benift of sun don't put carpet,cork or other insulators on floor where sun hit's floor. Without insulation sun will heat floor. If windows don't leak air heavy drapes is more bang for the buck compared to new multi-pane windows. The additional glass has no insulating value at all. The air trapped between panes is what stop's heat loss/gain. Heavy drapes create dead air space that does the same thing. Replacement windows are far overrated and expensive. To properly install a window require's extensive demolition which is rarly done. Botched installations are common. Before spending that kind of money,have someone you know has has experience in home construction and you TRUST look at yours. Caulking and weatherstrip repalcemnt can make a big difference for little expense.

I realize you are not pleased with some things previous owners did but I doupt they had anything to do with under slab ducts. Not saying it isn't done but ducts beneath slab aren't usually installed unless insulated. Before stopping them off try and have a trusted person have a look. I could give countless cases where ac techs made untrue statments. Not neccessarly being dishonest,they simply didn't know better. One come's to mind where ducts were underfoot similar to yours. When I went out to look at hvac for new owner,the neighbor came over while I was at outdoor unit. He said rest of house was always comfortable but the dining and living rooms had never heated or cooled well since house was new. What I found was carpet scrapes that had been thrown into floor openings. They had blowen down and piled up out of sight blocking air flow. You need someone with training and experience to help you make decisions and they need to be there,not in another state behind a keyboard. I realize that isn't solving your problem but hopfully you get a trusted friend to help you sort this out before spending your life savings chasing it.

yes, anything that shades the house makes a difference. you could remove the awning and replace it with a pergola that has a really open structure that lets lots of light in. you could install a fabric shade over it in the summer and take it down in the fall. (if your existing awning frame is nice, you might even be able to just remove the top and make a few modifications.)

there are also some pergola designs with louvers at the top that can open and close.

your HVAC guy was wrong about one thing: you do have options. you could switch to a ductless heat pump. you can control how much heat different zones of your house gets, and they're very energy efficient. they double as AC in the summer, and the good models don't need a backup heat source (we have a Mitsibushi Hyper Heat and it's amazing). it's a big upgrade, but a really good one if you can swing it.

Thank you everyone for the advice and help! I have a home warranty, I will check out too see if home warranty covers something with hvac to that room. That room is 69 degrees and the ac is set to 75. In winter the furthest part was 58 degrees and the heat was set to 72.