With ethernet (and hopefully MOCA) the speed of your 5GHz devices could approximately double, while the 2.4GHz-only devices should stay exactly the same. FWIW, Nest wifi users report the LAN connection of the first puck (that's the one connected to the modem on the WAN port) delivers ~900Mbps, while under optimal conditions the LAN ports of the other pucks in MESH backhaul mode are limited to <400Mbps actual (which is what would get halved when also using 5GHz for wifi). Wiring them in AP mode should make all of them see 900Mbps.
Even the AC2200 model Google Nests are still Wifi-4/N on 2.4GHz so would not improve things on single-stream 2.4GHz--they merely add one antenna for 2.4GHz and two for 5GHz in case you have the rare 3-stream or 4-stream clients to connect to them, which is unlikely. The latest AXE5400 Pro model does have faster 2.4GHz... but only if your 2.4GHz devices are Wifi-6/AX.
To convert a puck into AP mode, you go into the app under Advanced Network settings, Your Wifi Point and select Bridge Mode. It will then even work with other brand routers in case you wish to change to a different gateway router in the future (then you could repurpose all three pucks as APs). If you need more ethernet ports, then a switch can be wired to any of the LAN ports. That may be needed especially on the gateway, given you'd need a pair of MOCA devices for each AP so you would need to attach two of them to the gateway, one for each AP.
Alternating layers of copper and aluminum foil tapes on ethernet or coax cables can help prevent chewing, as with saliva the different metals will generate a mild electric shock, much less than chewing through a mains-voltage power cable would produce! But yes it would be best to remove the baseboard molding to put the wire behind it, or at least shield the cable with conduit or irrigation tubing clamped to the wall. For the cost of new MOCA devices and routers, you might even be able to pay an alarm installer to run the wires inside your walls.