I would not start demo-ing until you have a solid plan AND budget in place for what you're going to replace it with at a point TBD. It would take a lot of work and money to tear out what you have currently to get it to look like your inspiration pic. And if you don't have a plan and the money to do it, you're going to be living in your demo for a long time and that will not be enjoyable. And just a piece of advice .... you're on a very slippery slope if you don't put some boundaries in place now around where you let your kid play. Kitchen cabinets are not a child's space for a number of reasons, safety only being one of them, hygiene being another.

That kitchen won't appeal to very many people. Do you have plans to sell soon? I think people should design their home to their tastes, and to work for them. If they are selling, they shouldn't do anything more than clean.

Are you getting rid of the dishwasher completely? Planning to replace it with a new portable one? Do you have other counterspace in the kitchen? Room for an island/worktable? Have you looked at the prices for commercial sinks like this?

I don't think you've thought this through.

I would not go with your inspiration picture as there is no storage. You may not use it but a buyer may need it. The inspiration picture is nice but not realistic in my opinion. You are forced to be neat at all times or risk the look of clutter. In my opinion pictures such as those are very misleading to regular people who have more than one small appliance etc.

May I suggest you remove the stone backsplash and top. You can either do another stone or a laminate (Depending on your budget) and add a tile backsplash.

I would also de clutter the counter by using one of the base cabinets and adding a child proof lock so you can still have your little one explore.

New paint on the walls to add some color and fresh greenery and you are set.

I think first you need to know why you have mould. I happen to like the look of the freestanding sink etc but certainly not for everyone. Other than agood cleaning I would wait to affor a proper kichen makeover, those cabinets are not worth keeping and I certainly would not spend money on new counters and backsplash.

I think you need to plan for a full remodel. In the meantime, what have you tried to clean the mold with? Concrobium is a better killer of mold than bleach. After you have addressed that issue I would recaulk with new mold resistant caulk. If the countertop with backsplash is truly untreatably moldy then take it off and put laminate with a drop in sink as an intermediary step. A price point that is appropriate for those cabinets. Repair the cabinets if they need it. And save for a future remodel.

Not to point out the obvious, but mold is growing because those areas are damp and dark. Clean the heck out of them with bleach, let it dry a day (I know it’s tough with kids) and then caulk all the joints. No more mold! Any remaining spores are sealed in the wall and will go dormant as they need moisture to survive.

Your inspiration kitchen would not appeal to many, and will cost a bunch to get to from where you are currently.

Ripping it out will not devalue your house because you will not be able to sell it with mold there. The mold will have to be remediated first. Bleach will kill the mold, but it has to come in contact with all of it, which probably means ripping the cabinets out to get to it. And then you need to find the source of the moisture if the mold keeps coming back.

A couple easy things to try.

1- Clear out under the sink and take some photos of the underside countertop way back with the flash on. Aim for the area in the back where the faucet is. I'd bet that the problem is water dripping from the faucet and dishrack area down between the backsplash and counter. Another venue for a problem is a lack of caulk around the base of the faucet. Some faucets can't be installed tight to the countertop without a thin bead of caulk to keep water out. Hence, your photos may show what is going on.

2- For a quick test, tape some 3 inch wide strips of paper towel along the back edge where you are getting mold. If you see the paper towel getting wet, then that's at least one cause of the water infiltration. You'll need to clear off everything off the countertop for a day or two. If you use the dish rack, put a few fresh dry paper towels under it toward the back. If your countertop is slanted backwards, water will run from the dripping dishes toward the backsplash.

Are your counters stone, solid surface or laminate? If it’s laminate, that is a major issue with a under counter sink like you have. Even if stone or solid surface, it looks as if there is a large gap between the undermounted sink and the counter.

Hydrogen peroxide is a better killer of mold. It kills the roots; bleach doesn't.

Before you re-caulk, take a really small brush and 91% isopropyl alcohol (not the weaker rubbing alcohol), and scrub the grout. Alcohol dissolves the grout slowly, and you'll get down to a new clean surface after a bit of work.

My question is: How have I lived in this house for 17 years and never had a problem with mold? I'm not a super-cleaner, don't have a cleaning service, yet we have no problem with mold nor mildew. And we live in the humid mid-Atlantic region.

I would worry about your toddler playing underneath a moldy area. This is a health issue that requires your precious little one doesn’t get sick.

https://www.familyhandyman.com/cleaning/mold-and-mildew/how-to-remove-mold/view-all/

Take all the crap off the counter and stow away in lower cabinets if necessary with standard baby proof locks.

Easy maintenance starts with having relatively uncluttered surfaces so that you can wipe everything down without moving stuff. The clutter on your counter makes them a maintenance nightmare and I suspect that the back surfaces aren't cleaned frequently.

Once you have a clear surface, it would be easy to keep clean. Are you sure it is mold and not just standard gook that tends to build up wherever there is caulk or grout. My bathroom didn't have mold but the caulking between the counter and the wall mirror would become dark periodically and then I would have it replace. Similarly wherever there are seams in a kitchen, gook will eventually darken and have to be deep cleaned or replaced.

As for your inspiration idea, it's a terrible one and completely unappealing to most people. Take another look at all the clutter on your counter - as I wrote, it's not only an eyesore but adds to maintenance issues. I have no idea why you are not utilizing your storage which would simplify maintenance as well as providing you with a more streamlined aesthetic kitchen. Installing rollouts on your current lower cabinets would provide you with better storage and again, easier maintenance because being able to more easily access and stash items helps maintenance.

At some point, someone would probably want to remodel your current kitchen because it isn't that functional as it lacks drawers on lower cabinets.

Set your budget first! Do some investigative work about mold. Could be as bad as black mold behind counters and in walls. If that’s the case then big money to have removed.
Shop ikea for kitchens. Their kitchen furniture can me movable like in European homes. Some owners take their kitchens with them. That way if later you can invest more then station it all down and upgrade items.