The AirPort Extreme can handle up to 50 Wi-Fi devices and 100+ Ethernet devices. If you need more Ethernet ports you can connect an Ethernet switch to one of the LAN <--> Ethernet ports on the Airport Extreme.
Switches come in 5-port, 8-port, 16-port, 32-port etc versions and you can cascade or combine switches if needed.
A mesh system is going to provide more Wi-Fi signal coverage over a wider area since there will be multiple access points all providing a Wi-Fi signal. In other words, the mesh system in effect provides one "big" Wi-Fi network.
Ideally all the mesh devices will connect using Ethernet back to the "main" mesh router, although some systems use Wi-Fi to connect the mesh access points together. The wired Ethernet approach is far superior to the Wi-Fi approach, so you would want to connect mesh access points all together using Ethernet if at all possible.
We cannot recommend a mesh system because we do not use this type of system and we also only recommend products that we have used ourselves. Other users might have some suggestions.