No idea about the MAX, have only flown the classic and NG (fail operational). According to the FCOM for the NG the 737 is capable of continuing for any single failure below AH (200ft). That includes engine failure, hydraulic failure and so on. The only thing mentioned for the rollout phase was a flashing red A/P disengage warning or VOR/LOC and or G/S and A/P yellow indication, FCOM recommended action is "execute manual rollout".
There is an increased warning inhibition below AH, for example NO LAND 3 or LAND 2 as well as NO AUTOLAND are inhibited and have no consequence, the plane continues to land and rollout just fine. The required equipment table for CAT IIIb mentions that one needs 3 IRS instead of 2 (ISFD acts as third IRS and may be inoperative for CAT IIIB with a DH), that 2 engines are required to start the approach and that antiskid has to be operational. Those things would not be required for a CAT IIIa approach including manual rollout, although automatic rollout on one engine actually works perfectly fine.