Drivechains, extension blocks, and similar mechanisms are block size increases by another name. They offload transaction throughput, bringing cheaper fee costs to users. Would the base layer actually be worth more to miners? That depends if overall combined throughput increases enough to account for the cheaper fees. We simply don't know, and it's dangerous to rely on that. This is the block size and fee market debate all over again.
Currently the transaction fee is only a small percentage of the money that miners make, the rest are from rewards from mined blocks.
BIP-301 is one of the whitepapers for drivechain and the one that connects bitcoin's consensus to drivechains. By that I mean a miner needs to pay to mine drivechain blocks.
I believe the reason the transaction fees become smaller for users is that miners won't have to validate drivechain blocks or do PoW on them. Drivechains are trying to solve a hypothetical scenario where miners are keeping track of future validated blocks. I don't think this situation is actually happening in the wild because the vast majority of miners are owned by mining companies who would have to order the employees to make the miners send extension blocks to each other. So it's motivation seems futile to me, preparing for something that may not be happening.