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Change can definitely cause privacy issues. Both in Wasabi and Ginger, the wallet tries to avoid change by selecting coins in a combination that will eliminate the change or offers to "send a bit more" or "send a bit less" to avoid change. Donating also makes sense; I think it is possible right now manually. Or the third option is to use the small changes in a coinjoin transaction with the red coin option, which means that the wallet will only introduce one non-private coin per coinjoin round.
Payment batching is definitely something that people don't talk about too often. About the only times I hear about it are in the context of exchanges processing withdrawals.
I can see some scenarios where batch payments make sense, especially if it helps to remove the need for a change output. On that note, someone recently suggested to me that wallets should include a "send change to my favorite project" button and in conjunction with payment batching, I could see this being really useful. Batch my monthly payments to get them close to the whole value of a some combination of utxos I hold, and then donate the leftovers.
I also haven't paid much attention to Ginger Wallet. You make me think I should give it a look again.