There is no "TV version". The switch docks to a docking station to make it output to a TV. You just need a docking station and controllers to make a single Switch into a shared screen experience in the living room. Anyone's Switch can use the dock.

Physical games are sharable, but only one device can use that game at a time, because they're physical cartridges.

Personally, I'd go with a Steam Deck over a Switch, unless your family specifically is looking to play Switch games that are exclusive to it (which technically with emulation the Steam Deck can also play, but that's not legal unless you own a switch and the game). The nice thing about Steam games is that Steam's Families feature lets you share the entire game library digitally to 5 family members, so unless they want to play the same game at once, you only buy games once and they can all play them. There are also some games that let you own one copy and let multiple people play multiplayer at once on it, too.

Plus, games on Steam are cheaper than Switch games and the Steam Deck is only a bit more money upfront than a Switch is, especially on sale, which I expect it to be on sale for Black Friday coming up.

Finally, Steam games also can be played on a PC. Any PC. The Steam Deck is just an easy to use, skinned UI PC. As such, when the Steam Deck becomes obsolete, you don't have a bunch of games that are now locked to an obsolete platform. There are PC games that are decades old that still play on PCs today (although sometimes a bit of fiddling is required for REALLY old ones).

Edit:

FYI the regular switch and OLED can dock. The switch lite cannot.

Isn't it the same as with every other entertainment system? I grew up with a big brother and a little sister. We only had one PS1, later one X360. We could either play in co-op, or take turns. Sometimes my father would also play on the console, and we'd do something else in the meantime.

What's different about the Switch? It's an entertainment system. You insert the game, you play. I don't have one, but I'm pretty sure it allows for different accounts to be created and each have their own save file, so there's no need to buy multiple consoles/multiple copies of the same game. You can either play on the go, or hook it to the TV and play with the bigger screen. You are not forced to play party games just because you have a bigger screen, and you are not forced to treat it like a "personal device" just because you are playing on the smaller screen (I also despise the idea of "personal device" for kids: learning to share games is a very important lesson for kids).