In the context that there's the Ordinals Chain merge mined with the Bitcoin blockchain, doesn't the miner earn rewards from both chains, and therefore has the choice of which coin to sell to pay for electricity bills/other expenses?
It depends on the sidechain construction. If it is like NameCoin, which created its own tokens, out of thin air, then yes, this is just abusing 21 million coins limit, and then you have separate coins. But you can also have something like RSK, where all coins have 1:1 peg with BTC, and where nothing is created out of thin air.
Does merged mining add extra work for miners
If you mean "can the hashrate increase", then the answer is yes, because those miners are just mining regular Bitcoin blocks.
and therefore is like a block size increase?
Not really, unless you make it mandatory for a full node to follow that chain, but then, you need a successful soft-fork to do that. But in case of additional data, there are no new consensus rules behind that. Those data are just huge OP_NOPs, which means, no additional Script needs to be enforced. Which means, there is no need for full nodes to follow that data, and download it. They only stop at verifying signatures, all data behind them are posted separately, and are not processed by full nodes, because it is not needed to determine, if a transaction is valid or not.
If there was no incentive to mine the Ordinal Chain, then why try to find blocks there at all?
The incentive is the same as with mining Bitcoin. Which means, you can mine Bitcoin only, or mine Bitcoin, and your own chain, at the same time.
Note that every time, when you create a transaction, you can commit any data into each of your signature, without increasing the size of your transaction.
Plus it would be better for the miners if Ordinals inscriptions stay in the Bitcoin blockchain/keep fees high.
Miners can accept fees as they please. If miners would want to increase minimal fees into 1000 satoshis per virtual byte, then they could just apply those rules, and not include transactions into blocks, if fees are insufficient. Or, the biggest N pools could just secretly agree to flood mempools with their own transactions, just to increase minimal fees (because then, they will get those fees back in their own blocks they mine).
fees would still be this high and worse if 1% of the population started using BTC on chain for daily payments
That's why the solution is to handle N users per coin. Which means, if you have 0.01 BTC transaction fee, and there is one user behind that, then it is quite high cost. But if there are 1000 users behind transaction of the same size, then each of them has to pay only 1000 satoshis.
And because batching N users behind N-of-N multisig is easier than combining N ordinals (because if you combine signatures, then their size is not increased), it should be sufficient to give regular users a tool to compete with Ordinals.