If you are eliminating radiators because you don't like them, then you won't like radiant floors. The experience is similar. Steady warm heat that heats the objects in the room to a steady state. If you'd keep the radiators, you possibly could use the same boiler to power the floors. Possibly. Otherwise, a mixed system is the most expensive and least comfortable choice. Pick one type. And I'd personally excavate for a basement. In for a penny, in for a pound. Pier and beam in a cold climate is not common. For a reason.

Thank you @The Cooks Kitchen! It's not that we don't like radiant heat- it's that we don't like the space the radiators take up as we have a very tiny house as it is. We also don't like the ductless AC options. And, yes, we have an excellent boiler that could be used for the radiant floors, so it won't be an added cost.

And your point is well taken about the basement, but the extra cost was just not feasible for us especially considering our house is a semi-detached and not possible to excavate on our neighbour's side.

Does your boiler also provide your domestic hot water? Keep in mind that if your existing boiler is sized to provide full-house heat, and the extension also will have ducting for A/C and heat, that boiler is way over-sized and is going to be operating very little, i.e. short-cycling. It won't be efficient and it will be hard on it (less so if it's providing hot water, but still). Just to provide slightly warmer floors in a house that's already heated.

I don't like any of the options but to throw one more into the mix - you might consider in-floor dry (electric) heat. It will cost less to install, but more to operate. However if you're only operating it infrequently to take the edge off the coolness of the floors, particularly in the coldest weather, that may not be too expensive. Assuming you have time-of-use rates, you could arrange to program it to only operate at night. I'm not saying this is ideal but perhaps worth considering.

It may be the least-bad trade-off in terms of modest up-front expense, avoiding the issues with a mixed system, and making sure you don't finish a big reno with that one regret and annoyance that floors are colder than you'd like. Doing something to fix after the fact will be expensive and a hassle you will not want. (There will of course be some other regret)).

(There are other options like radiant floors throughout the house or less intrusive radiators, but assume you've already considered, and that's before considering A/C needs)

The floor of the new addition would feel cold if it were being built on a concrete slab sitting on the ground. In a cold climate like yours the slab acts like a giant heat sink. In that case radiant heat would be a nice option. However in your case the addition is sitting on a pier. You can add insulation and keep the floors warm without spending additional money on radiant heat and concrete. I think installing spray foam insulation underneath the floor would work well in this case.

If you don't install the radiant heat, then I don't see why the old and new floors can't be at the same level. Unless you like the sunken floor look I recommend keeping the floor the same height.

If there is not radiant then you really have no use for your old boiler. You could integrate into a forced hot air system but I think the efficiency would suffer. I also think you are better off getting a stand alone tank hot water heater. You can also get a tankless type if you need to run multiple showers in a short period of time.

As nice as it sounds to have heated floors in a cold climate, I worry that the slab over pillars will make this VERY difficult to achieve (ie....very expensive). To do this properly, you have to incorporate a HEAT BREAK into the slab. Yes. INTO the slab.

Example: if this were on the ground, your HIGHLY EDUCATED slab expert (who is fully trained in thermodynamics...ahem...fully trained) would design a system where the heat block was installed FIRST (sitting on the ground) and then the concrete poured OVER TOP. As in closed cell polystyrene blocks (several inches thick). The foam blocks would STOP the transfer of heat DOWN into the earth (nothing like trying to heat the GTA through your heating system!).

When working with pillars, this gets complicated because you STILL require the heat break. But WHERE to put it. Now you have to put it ON TOP of the slab. Then you have to use something sitting ON TOP of it to prevent it from melting when exposed to the heating system/coils/tubing.

There are multiple systems that sit on top. They are fast, easy but expensive because they are fast and easy. They can/will raise the floor height by several inches....and that has to be incorporated into the design because the STAIRS will be pushed out of code.

And once that heating system is in place you have to be aware that your FLOORING choices have just been GREATLY reduced. Example, that beautiful hardwood you have chosen MAY NOT be allowed over ANY form of radiant heat....at all.

Whew! It is SOOOO much cheaper to get rid of the old boiler and upgrade the HVAC system. Much, much, much cheaper. And the operating costs will be MUCH lower...because old systems are usually inefficient.