If you put your router into "modem only" mode, you'll have no router/NAT/firewall, so you'd have to add another router - so you don't want to do that.
Wi-Fi and "routers" are unrelated - it just happens that in a SOHO get-you-on-the-internet omni-box, Wi-Fi is included as part of the package. "Modem mode" versus "routing" mode is nothing to do with Wi-Fi, it's about which device you want to have do your routing. However, one of the consequences of modem mode is that it (probably) defeats Wi-Fi (and all but one of the ethernet ports,) but defeating Wi-Fi is a "consequence" of modem mode, not the "purpose" for doing it. A lot of SOHO routers will let you simply just turn Wi-Fi "off" if you don't want it.
These new fangled "mesh" systems are integrated solutions, so you can't just buy any AP and use it with any other AP - they will need to be able to "talk" to each other and in the SOHO realm that almost inevitably means you need to buy all your AP's from the same manufacturer, possibly from the same product line or as "kits" or "packs." (This is nothing new by the way, enterprise scale Wi-Fi kit has been like this pretty much forever.)
The way the term "mesh" is being used in the current nonsense-du-jour is generally to mean "uses Wi-Fi for the backhaul links between the AP's." If you have UTP (cable) to all the locales you want to deploy extra AP's, you'd be much better off to use UTP cables for the backhauls - it'll be faster and more reliable.
There could still be advantages to a "whole home/mesh" type system in that the AP's might "talk" to each other to do things like aid the aforementioned hand off between AP's, agree a (radio) channel plan and so forth, but I suggest using radio for the backhaul data links is a retrograde step if you can do it by UTP cable instead. Of course, "mesh" AP's that can use either Wi-Fi or cables for the backhauls is the best of both worlds.
When you get into planning multi-cell Wi-Fi deployments (however the backhauls are achieved) beware of falling for "Big Wi-Fi Myth Number 2" that client devices are "always hunting for the best signal." They do not. When to "roam" from one AP to another is a decision taken by the client device not "the system" and some clients needs things to get pretty grotty before they initiate a roaming assessment. You also need the SSID's & passphrase to all be the same - if they differ the clients will never roam.
Recently there are some tweaks in the Wi-Fi protocols so that systems can send a kind of "hint" to a client that it might do better with another AP. Whether clients "take the hint" depends on whether they are "hint compliant" and of course, many older clients are not. Also of course, the infrastructure of AP's need to be smart enough to determine what hints to send to who. Again, this used to be the preserve of some (but not all) enterprise systems and possibly this is being included in these "whole home" systems, but you'd have to check the specs carefully and shouldn't "just assume" it's included just because the marketing people are calling it a "mesh" system. There are other ways to achieve similar results using "less friendly" mechanisms, but I doubt anyone would be selling kit that does so as it's not very "user friendly."