Is it worth buying a used video game console?

Usually yes: consoles are built to last, and a tested used one costs far less than new. The risk is in the details: worn controllers, a dead disc drive, missing cables, or a retro console that needs an adapter for today's TVs.

Check before you pay

Tick them off as you go:

The listing says "tested", ideally with a photo or a video of a game running.
Controllers included and working; on modern controllers, check that the sticks do not move on their own (drift).
The disc drive or the cartridge slot reads games.
The right cables: older consoles plug in with composite (yellow, red and white) or RF cables, and many recent TVs need an adapter for them.
For a recent console: it signs in and goes online, and the previous owner's account has been removed.

Red flags

"Powers on" as the only test.
No power supply with a console that uses its own special one.
Photos that hide the controller ports or the disc slot.

Which ones still make sense

Nintendo Wii, PlayStation 2 and the Nintendo DS family: huge game libraries, and cheap.
Nintendo Switch and PlayStation 4: current games; check the controllers and the account.
The original NES (model NES-001) and other retro consoles: buy them tested, the cartridge connector wears.

Live listings on eBay

A few current listings, United States. Asking prices; read each listing's condition and returns.

Sources: Console model numbers on the label underneath; general care and connection guidance.