Then I would probably just setup regtest and mine in there. (not sure if this is possible but I think you can mine on regtest)
It is possible even if you use CPU for mining, but when I tried it, I had a lot of stale blocks, because the difficulty was too easy and mining was faster than sending blocks to the clients.
Can't you just increase the difficulty? Actually, it should self-adjust in Bitcoin, after 2016 blocks.
Is there any good reason why so many miners are apparently mining it?
They must have an incentive. I can't really believe that paying the hosting of a faucet website plus the mining costs are less than the ad revenue, but they must have a profit somehow, otherwise they wouldn't mine. That profit may not be translated into money.
My best guess is that testnet is used to test custom mining software, and any improvements and/or bugfixes to mining software. It may also be used to test things intended to reduce orphaned blocks of the miner.
Okay, this makes the most sense to me right now! I imagine Bitmain and the other ASIC manufacturers all pre-test their hardware for a bit of time before shipping it. Especially during the development of new generation miners, they might not want to use mainnet directly, so they point those beasts towards the testnet. Sounds plausible!