But isn't merge-mining going to be like a block size increase? Because it would require miners to use more processing power, which will also require them to spend more capital to mine.
The difference to a block size increase is that only miners would really be affected. It would still be cheap to operate a full node
without mining. With all advantages this provides, for example privacy, and helping against other attacks on SPV clients.
Transaction processing costs on Drivechains or other merged-mined sidechains/altchains, in relation to hashing costs should be negligible for miners. Miners are generally extremely well connected to the network, so the biggest problem for "household" full nodes, bandwidth, does not really bother them. And historically a big percentage of the Bitcoin miners for example merge mined Namecoin, even if the NMC block reward was/is a small fraction of the Bitcoin block reward. It's ... at least something.
You could bet on that if the Sidechain/Drivechain model gets popular there will be more than one sidechain. Let's say we have our 4MB/10min
BTC mainchain and 10 4MB/10min drivechains. Thus for a full node operator who wants to use one of these chains and in addition the mainchain, they still would have much lower transaction processing and bandwidth costs (8MB/10min) than if Bitcoin had 44 MB blocks to accomodate the same number of transactions.