Good computer for new gaming at lower price?

[Silfir]

Since you're looking for a gaming PC, onboard graphics should not be relevant to the conversation unless your budget is like $400 or something. (In which case a Ryzen onboard chip outperforms Intel's, as far as I'm aware.) $1000 should get you a machine that can handle anything. Think RX 580 or GTX 1060 at the very least; GTX 1070 if you're lucky.

There isn't really such a thing as a go-to prebuilt PC. Sometimes your best option might be to go with an older, discounted model (typically an Intel i5i7 from a couple of generations ago - avoid any AMD CPUs not named Ryzen like the plague) that can easily be upgraded, which is still less of a hassle than building one from scratch (and can sometimes be more cost-effective).

I'd still build one myself. Your kids can help.