If you are somewhat tech savvy start here: OpenWRT

I don’t buy the latest flagship routers so I’ve never had to set up one without having to create an online account yet. I bet if you look at some Netgear (or similar) models from a couple of years back, they won’t have online activation and they just have the default admin credentials on a sticker under the router.

If you want a more up-to-date one, GL.Inet routers ship with OpenWRT installed, and you don’t need an online account to activate them.

If you get a cheap n100 or similar style mini PC with multiple network cards and install pfsense on it, the only account needed is the local login. It took me a lot of YouTube tutorials to get it working just the way I wanted, but it's a great solution.

Not quite clear to me, what you actually need. Only WiFi - then you'd need an access point, not a router. A router wold have something on the other end too, like fiber or ADSL.

Is there any way to get a new router to work without connecting an account to it

Yes. Look at Mikrotik offerings. You'll need to have an understanding of networking, it's not plug and play. But it's fully self contained and powerful. No external accounts needed, just a local/device admin account and password.

I've been thinking about GL.iNet routers. They have OpenWRT-based firmware, and it seems typically can also run vanilla OpenWRT.

Just found them on proxysto.re when looking at physical Mullvad vouchers, and regretting getting that on Amazon for (slightly) more money and with slower shipping.

PCWRT is OpenWRT on easy mode.

Build your own

I use pfSense with eero access points.

I don't know what are u talking about but used Asus tplink routers for home WiFi all it wanted activate login admin password