Re:
I had the 2012 nexus 7
Great thing, technically I still do have it but a visit too the floor killed the screen. My fault and one of them smallest of drops too. Its taken much more before with the kids.
Kids games work well as well as the cbeebies app.
Web browsing is fine given you have whatever browser you want and always the latest Chrome.
As mentioned refurbs are less than 100 quid on these. Try get the 32 over the 16 if it for the family.
You can each have your own account and the kids a restricted account if you want.
On any android device also download the Amazon app store as you can get loads off the free app of the day.
But try them out, not sure what the 10" devices are at but I've got a Toshiba Exits Pro. A silly high res device with a high speed processor in it. For general use I would recommend it but for hard processor use I wouldn't, while extremely fast it needs to throttle and they've not done it as well as they could. Powering the silly high res screen doesn't help it.
Kids and wife love it though. Was only 180quid or something of Amazon in a deal. Hukd is a often nice place to see current offers on these things.
Still go play on them, I pads do just work and you know what you're getting. You're just tied into them a bit more and no chance of budget devices.