According to what you said, it seems that right now my best option is probably to buy Time Capsule from eBay.
Not sure about that. All of them are a few years old at least. And Catalina is proving very unreliable to the TC.. there are alternative backup software like Carbon Copy Cloner which will do a better job.
Anyway do you already have a mesh setup?? .. if not make sure you buy one with gigabit ethernet ports. e.g. the new google nest wifi has no ethernet on the satellites. Make sure you have the possibility in the future of used wired connections for backhaul.
You said something about Access Points, is that the way?
Access Point are wired by ethernet back to the main router. If you are prepared to invest in wiring, at least one outlet on each floor of your home, the result will be far superior to mesh.
Let me make clear there are a few different grades of equipment here.
Older Extend wireless systems like Apple used with airports is slow.. at least halves the connection speed and big increase in latency. And has poor range and no possibility of multiple hops.
Cheap mesh systems of which the Google wifi or TP-Link Deco series (M5) are good examples.. use 866Mbps links for both connecting clients and backhaul so are easily saturated. Better than extend wireless but still cause slow down and increased latency. Mesh gives superior flexibility and roaming at least sort of works.
Expensive mesh systems like Velop which Apple sell in their stores and Netgear Orbi use a third wireless on high speed 5ghz channel as purely backhaul. So they usually come out tops in most comparisons of mesh systems.
https://www.tomsguide.com/us/netgear-orbi,review-4263.html
Better they can be used with ethernet backhaul which improves speed and latency over even the best wireless. Some of the cheaper ones also allow this.
When used with ethernet your normal Asus and Synology routers also work well in mesh or AP mode. These different modes of operation are nearly identical only mesh removes all control from end user whereas AP mode allows you to actually connect onto the unit and set different wireless for example. Roaming should work well either way if setup for it.
Where you need to be careful is trying to mix and match. If you go for a particular mesh system for example.. you MUST stick to that brand. It is not interchangeable.
So can you plug a TC into a mesh satellite or main unit??.. answer is yes as long as it has ethernet ports.
Will it be reliable? Hmmm.. hard to say. Not great.. but there are some things in the setup that help loads. Nothing can quite guarantee the results.. and ESPECIALLY with Catalina which can fail miserably even in pure Apple setup.
If you go down this path.. be aware it is fraught.
I personally have it setup like this.. Asus main router. TC plugged in to switch in the network. Multiple AP units around the house. Reliability is fair with every tweak I know. I would not however recommend it to anyone without decent DIY and network skills.
I also have a synology NAS on the network and that is so far 100% reliable. A lot more expensive but you gets what you pays for.