Hi @Kramek,
Just some thoughts as they arrive in my brain.
Have fun thinking about this and asking questions and learning....but the medical must not be overlooked or rushed. Given the apnea, asthma and meds you listed your medical will be complicated.
Buying a plane with the purpose to learn in it and save money really won't save you that much in the end. As an owner you have insurance, hangar, finance fees, mx reserves, annual, etc. All that as well as fuel will probably get close to a Archer or 172 rental rate. I just learned in our 182 and with all costs accounted for i probably paid close to 75% of a rental rate.
Because you want to haul 6 people around, you're going to blowing thru lots of fuel and mx if you get one of the 6-seaters you listed. Now, If you already had lots of flight hours with other pilots, know 100% sure you love it and wanted something a bit smaller and more common then buy it and learn in it. But with your desire for a 6-seat plane, at least get your PPL in a smaller rental first.
Don't forget about clubs. There's one near here with (2) Centurions
I've never done W&B on a 210 but while your family is young it might also work.
Probability of all 6 family members being comfortable with flying might be a risk area. Pretty much all the singles you've listed can get bounced around noticeably on hot summer days, seats are small, it can be smelly from fuel, no potty, no a/c, etc. It is not like commercial (but can be fun!!!)
Probability of making all those trips each year, especially the first few years as you gain confidence in wx - will probably be 50% in the summer and like 20% in the winter. And there will be times you will leave a day late for wx or return a day early for wx. So let's say you wanted to do (6) 3-day trips with the family. Until you are a IFR, very confident pilot you'd probably get 3 of the 6 flights in and at least one would be cut short.
You mentioned finding a CFI with their own plane. That doesn't seem very common. Maybe I have this wrong, but if they use their own plane and charge for flight time I believe it is then a business and no young CFI wanting to make the majors is gonna mess with that due to other requirements?
You say it will be cheaper to fly your plane with all 6 vs airlines. I get that and if you did it enough you'd be right. But just remember flying will cost way more than driving which is what you are used to now. Where driving might cost you $50 each way, flying might cost you $200 each wsy.
Given the complexities of 6 seats and some of the odds above maybe you will wind up with a nice 4 seater like a Dakota and a few of you fly while the rest drive. A compromise.
Now if I was asked to fly 5 people and me, the first thing I'd do is hire a cfi and a trip to the gas station to buy rock salt bags to simulate those precious souls weights and do several flights to have a good feeling for the CG, takeoff, climb and landing characteristics - especially in summer.
Flying is amazing! Get that medical figured out and take a discovery flight while that's going through. After that knock out your PPL. By the time you are done with that and fly for a year a lot of the things being said here by the others will have more meaning. Your optimism will be tempered by experience.