I can. I have. I considered lemons to be a very cost-effective way to get competitive driving time. The last race I participated in had the top 20 teams of 107 within 10% of the laps of the leader after 14 hours, with the top 3 teams only a few laps apart. Hell, one of the races I drove in saw 1st and 2nd split by *when* they finished their last lap, and that's after 14 hours of racing. Not 10-20 minutes, like a Kart race. To state that Lemons/junker racing isn't competitive is laughable and only serves to demonstrate your ignorance.
For my first real race, in Chump Car, I bought $1k in safety gear, paid a $750 drivers fee on the team, bought a $100 competition license, showed up at the track and got 1 hour of practice time and 4 hours of racing time. So, for less than the cost OP has budgeted for the whole car I got 5 hours of seat time going wheel-to-wheel with ~100 other cars (not 3-15 like Kart). I spent little time prepping the car because that burden is spread across the team. I didn't have to work out the logistics of transporting anything other than myself. My fee covered fuel, brakes, tires, etc. After the race I am free and clear of maintenance costs on the vehicle, the team principals manage that and wrap everything into the driver's fee.
SFI safety gear is good for many years, so I've been able to drive in several other races for simply the drivers' fee and annual competition license. All said and done I've paid about $5,200 for around 24 hours of wheel-to-wheel racing time and had a blast doing it.
Equivalent time in Autox works out to costing $50/8min*24hr=$9k + helment + tires + fuel + brakes
Equivalent time in HPDE works out to $200/80min*24hours = $3.6k + many sets of tires, brakes, tanks of gas, and wear+tear
Karts cost on the order of $5k -12k depending on what tickles your fancy (as an unassembled kit cost), plus the same $1k in safety gear, plus track fees. Even if the Kart fees are significantly lower than full-size cars you'd have to do a lot of driving over a lot of years to see a return on investment per wheel time. It also appears that you don't get a ton of drive time per weekend, given the 10-20 minute length of most Kart stints, with 3-4 heats + 1 race.
And, frankly, if I'm going to drive at reckless speeds against other lunatics like myself I'd like to be inside of a metal box, not on top of it.