First off, you don't need 250+ hp to have a seriously fun car.
- Agreed. You have to create your own fun. Living in the Midwest means I don't get windy roads nor canyons to play in so a lower HP, but great handling car doesn't have many places to shine. The only way to play with a cars performance is in the occasional roundabout or merging onto the off ramp/backroad.
Slow car fast.
Do you want to have fun at the track only or have fun all the time? Lower power lets you run through the gears on most streets and on ramps without really breaking the speed limit (or by much). It's a ton of fun on the street street or track because you're working the car for all it's worth, the car isn't doing everything for you. On the other hand, most high powered cars (Corvette/Ferrari/Lambo/McClaren/911) is absolutely boring as heck at anything below "go directly to jail" speeds (or at risk of taking out pedestrians) and if you do take those to the track you're going to see most of them getting lapped by the cars I just mentioned. Why? It's too much car for most people, especially without training even on a track. Corvette has the lowest insurance for high performance cars, why? Because they rarely are driven fast and at normal speeds the car isn't even trying, these days even a Camry is boring at city speeds with more than enough power to get you into trouble. This was the whole concept behind the Toyota Subaru mashup, not a ton of power, but a small fun to drive car.
-The cheapest BRZ I can find on CarGurus starts at 12k, but would be a solid platform. BRZs have such a cult tax associated with them. I fear they will be in the same vein as S2k, 240x, etc cars that just command a price because of what they "could be" vs what they are.
As someone who's been in a street/track prepped Miata, they're anything but slow or boring especially on the street or in trained hands. SCCA racers (Sports Car Club of America) were well known to drive early (new style) Minis when not racing because of the handling, most owned a Miata at some point as well (Mini was more practical). Go look at SCORE (Baja 1000, Mint, etc), they have a stock VW (vintage) bug class and those guys have an absolute blast racing (I use the term loosely) through the desert. There's also Lemons racing and a few spinoffs where the car has to be under X amount of dollars, usually a few grand and the whole point is to have fun with cheap cars.
- The Miata is a track weapon, but not something that can be a every season driver in the great white north, doubly so as a parent. (NC Miata is best Miata IMO)
As for some being projects...
Every hobby is a project and while you can get many of these in pretty good condition for a decent price if you're going to do seriously hard driving (track time or not), pretty much ALL cars become projects. You're going to destroy tires, brakes, driveshafts, hit things, spin out, maybe wind up in a ditch and probably want more power/brakes/handling/traction/lighting/body kits, etc...
This is under the assumption of racing. Living in the midwest there are limited racing options. Dirt track being the most common and accessible, but there is a drift series that runs not too far away, but that's a whole other can of worms. I was speaking more in the term of a project car being the Sunday driver, nice weather car. Growing up I had a buddy whose dad had a Chevelle they bought in their late teens and has kept it to this day and only drives it when it's nice. Half the year we are covered in a mix of salt, sand, and ice so any vehicle driven during those 6 months are being sacrificed to the rust gods.
I'll give you that once purchased the computer is cheap to maintain and you can drive (almost) anywhere(?), but it too suffers the same fate, you're going to want bigger screens, faster CPU/GPU, newer games, updated pedals/wheel, keyboard, etc... It's not just a one and done either but it isn't a complete experience either, you're holed up in a room, not actually moving, not interacting with people face to face and not actually interacting with your surroundings and nature. You can sim drive the Tail of the Dragon as fast as you want (and not get a ticket!) but you didn't actually drive tail of the Dragon did you? You can sim drive Laguna Seca, but you didn't actually drive Laguna Seca did you. You never even left your room. To each their own, I'm not knocking the guy, just saying go touch grass.
-Very good point. I have looked a few times into getting into VR sim racing and my conclusion was I need around $1,000+ in sim racing hardware, another grand in VR, and more on how deep do you want to go. I use my desk for work so I cannot permanently leave my rig in place, unless I got cockpit, which then adds to the cost and footprint in my home. For that price I can put that money into my IRL car to enjoy it more (looking at you downpipe and a tune).
Any 3 series BMW, late 90's or 2000's would be your best bet, maybe a Z3. Mazda Miata, Mini, Dodge Charger/Challenger (look out for under serviced engines), Ford Mustang (look out for pedestrians), Chevy Camaro. All of these can be had in decent shape for well under 10k and easily sourced. You can occasionally even find a nice though high mileage S2000 in this range. Some have well over 250hp some do not, some will kill you on insurance (Dodge), some are reliable (Miata), some literally eat parts (BMW) but all are known to be a lot of fun.
-I was at one point looking at Z3s because there are like 3-4 2 seat coupes that are attainable for most people (Miata, Z3, MR2, and maybe a Boxter. I could be missing some). The only other car that catches my eye is a post facelift SN95 Mustang because that is one of the best looking cars ever. When I was looking for something fun with 4 doors (kids) and AWD (cuz snow), I was looking at Bajas, but that's another car that has a cult tax and decent examples start over 10k for a auto/turbo and if you can find a manual turbo you can expect closer to 15k unless it has mega miles
We are sadly in a much different era of project car than we were 5-10 years ago. I remember in 2015-2016 looking at stateside, imported R32 GTRs for 16-20k. Now you can't even look at an R32 GTR for under 30 and most are closer to 35. Hell, I looked into getting a GTST and doing an LS swap and those went from 10k to closer to 20k.