First, your question is not related to Apple per se. The functionality of your Linksys router is as Linksys offers that for this specific model. Users of other routers may have done something similar, but their configuration and how to set it up will be different, and may or may not involve a different primary home network. The best advise for your use would come from Linksys themselves or from Linksys user groups.
The router of your mesh system should do the DHCP, and your Linksys router should be configured to do no routing whatsoever (else you get a second network, interfering with the existing one), and get an IP address from the mesh system through DHCP. I see no reason why DHCP couldn’t work for this. NB: In this scenario, the Linksys router shouldn’t do DHCP itself: it’s own DHCP-server function should be disabled, but that should be automatic for bridge mode and no routing. If connected by ethernet, then use Bridge Mode.
A client-less disk can be connected to this Linksys router.
On the other hand, a new dedicated NAS would be one device, and have less points of failure, and thus be more reliable. I wouldn’t advise your idea to people thinking about getting a NAS. Even less so if they are self-proclaimed “not that tech savvy”. It will get issues at some point, and that requires knowledgeable troubleshooting.
