A mesh network is a last-resort because it wastes an awful lot of time shuffling data around a slow, unreliable, shared medium. Adding "mesh" access points, particularly if you are in a crowded WiFi environment is just shouting louder at a party.
Think about where you need to use the WiFi. Run a cable to the nearest convenient point. Stick an Access Point there. Switch the one in your router off if it causes problems, or do as I did and buy a decent router without WiFi built-in. I haven't lived in a house in the last 50 years where the ideal place for the modem/router (as close as possible to the telephone master socket) is also the ideal place for a WiFi access point.
You absolutely can run several access points with the same SSID. You don't need special kit and your phone or laptop will connect to one or the other, and swap when it gets a better signal, but unless they are a good distance apart don't, for heaven's sake, run them on the same channel(s). Even if it means confining them to (for example on 2.4GHz) "g" speeds rather than "n", you will get a better experience from two non-interfering access points on a "slow" standard than from two overlapping ones on a "fast" standard.
"Auto" channel settings on ISP-supplied router/access points are the bane of my life around here. Every blasted house is using channels 1, 6 and 11 - often all three because of "mesh" or "repeater" products - when almost the entire globe other than the US would be better using 1, 5, 9 and 13. It makes it difficult to slot my own APs "in between" weaker neighbours. For example, if I had a strong neighbour on 1 and weaker neighbours on 6 and 11 I could look at using channel 7 or 8, which would only partly overlap with the weak neighbours) meaning that whereas in the early days I could cover the entire house adequately with a single AP, I now need two, and frequently channel 13 is the only relatively safe bet.
If I want to cover the entire house, that is.
And why is it that even the radios which can use 5GHz rarely use more than the first four channels? That makes 5GHz little better than 2.4GHz, particularly when some twit has it set to "80MHz ac mode" and blatts out the entire band. How hard can it be to implement the interference-avoidance that is mandated for the other eight channels?
Oh right. It's just easier and cheaper not to.