The Switch is essentially that. Obviously not $60 but it's a several generations old Nvidia Tegra SoC. The AppleTV wasn't far off either before its massive price hikes.

Not $60, but a bit close. I'd do something based on the RK3588 SoC. It's pretty beefy. GPU isn't stellar but it does have good video codec, a proper PCIe bus including support for e.g. m.2 slot nVME on many boards, and decent amounts of RAM, and good a multicore basis.

And there's some amount of kernel upstreaming that makes it a bit friendlier than some.