Could I get away with this?

ISP coax --> ISP’s modem --> (need a router recommendation) --> random POE switch with enough ports to support all my gear? I might also add a UPS and surge protector.

Yes. Specific router recommendations will depend largely on your ISP speed. Your existing Google mesh router is likely sufficient. (not sure which unit you have exactly)

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Practically speaking, how would I initially configure/set up everything if I piece meal it together with various OEM gear? Let’s say ISP modem (should I get my own?), TPlink router, Netgear POE switch? Sorry if that’s a dumb question.

ISP modem is fine. Buying your own will likely save you money in less than a year, but you lose some of the “it is the ISP’s problem to fix” if things go wrong. That being said, it is rarely the modem that goes wrong. Really, any modem compatible with your ISP’s service would be fine here.

TPLink router. These, or really any consumer router would likely be fine. Again, it depends on how fast your ISP service is which model you’d need specifically. Also again, your existing Google mesh equipment is likely sufficient. Consumer level stuff will basically configure itself. It sounds like you’ve already configured your mesh system. It won’t be much different here.

Netgear POE switch. Totally fine. This shouldn’t need any configuration at all. You can get a managed switch and configure all sorts of things, but there’s likely little benefit to you to doing so and you’d need to get a router that supports VLANs and such to go with that.

Similarly, how much more difficult/easy would a similar Unifi solution be?

I can’t comment on this. I’ve never messed with Unifi gear.

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Overall, don’t overthink it. Modem plugs into router WAN port. Switch plugs into router LAN port. Everything else plugs into the other switch ports. Make sure you plug each of your mesh APs into the switch as well and configure them as hardwired units instead of using wireless backhaul.

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You can do this.

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Edit: formatting

As others noted, get a switch to put behind your TV. Plug your 15m cable, the TV, and the PS4 into that.

Later on, it doesn’t matter where you put the modem end of that 15m cable. Theoretically, you could have faster internet access for the TV and PS4 by leaving it plugged into the modem. Fewer devices in the communication chain typically equals faster connections. Practically, it won’t matter much. We’re talking fractions of a millisecond latency difference. There are other situations where leaving it in the modem is beneficial including if you have a faster connection to your ISP than you have for your internal network connections or if you have a lot of contention for bandwidth plugged into a single port on the modem. Again, practically, none of these will matter for you. Just plug it in wherever is convenient or wherever makes you happiest.

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P.S. That modem is most likely a modem/router/wifi access point combination device. Technically a modem is what converts from whatever medium your ISP uses (cable, DSL) to ethernet and a router is what allows you to use multiple devices with a single public IP. I assume you know what an access point is. If you have a fiber connection, it isn’t even a modem, it is something else. Again, none of this really matters practically for your setup.