@mntmn I had a similar issue when bringing up a board with a SD 3.0 interface; I don't remember the exact reason, I think it had to do with the protocol upgrade to 3.0 after the autonegotiation (something about 3.0 was not implemented correctly on the H/W or SoC config, maybe some voltage or enable signal missing)
eMMC is simpler than SD due to less circuitry in between, so there should be less stuff that can go wrong...
one debug step could be to disable the faster I/F versions and speeds?