If you tried to start a fork built off of older blocks with a bunch of your miner buddies, it would essentially be a 51% attack. You’d have to convince a quorum of miners and users to treat ledger entries in your fork like real Bitcoin transactions. If you have the power to do that, you have successfully attacked the network, and you may as well have just sat your new blocks with mining rewards on top of the extant main chain. It’s not like mining older blocks gives you more BTC in some mathematical sense - you’re always just earning whatever you think the network will let you get away with, based on the schedule that has been agreed upon a priori.
Hopefully this makes it clear that the question of whether a fork has a separate supply is confused. Neither the main nor the fork have a true finite supply - just a finite amount people will in practice agree to letting you mine. In the case of the fork, it doesnt’t matter whether you can convince them to let you go back to increased mining reward, because you have a much larger problem - nobody considers any transaction in the fork money in the first place. If at any time they do, you’ve attacked the network. Miners and users both equally dislike a state of the world where there is more than 1 version of history - it calls into question whether any wealth they have in chain A is real now that chain B is more popular. In practice they will quickly coalesce into considering one chain the true one, and changing everybody’s mind, especially long after the transactions have settled, would be a monumental task. Because everybody understands this, nobody would even try it, since it’s a waste of precious mining resources. This is why you didn’t see rogue forks forming when the reward decreased from 50 to 25 BTC.
I know that’s a wall of text, but feel free to keep asking questions if you’re interested.