With your requirements, nothing. There are no Android emulators for the original Xbox, PS2 emulation is still iffy at best on the highest end SD SoCs, and Wii/GC emulation, while better than PS2, still has issues as well with compatibility and performance. Your best bet if Android is a requirement for whatever reason would simply be to buy a flagship Android phone with whatever Qualcomms latest Snapdragon is and a controller. You can grab a used Samsung Galaxy S22 and a controller for like $350, which will likely have better performance than any similarly priced Android handheld plus a nice OLED screen.
Your best bet is an x86 based handheld running Windows or some flavor of Linux instead. 6th gen console emulators are much, much more mature and stable on x86 platforms than mobile, so you'll get better stability, support, and compatibility going this route than any Android handheld.
The cheapest option will be the Steam Deck, which should handle the vast majority of Wii/GC/PS2 just fine, but doesn't have an OLED screen, just IPS. You can get away with buying the $400 64GB and then picking up a 1TB 2230 NVMe for another $100 for less than the 256GB model.
You could also go for the Asus ROG Ally, which has slightly better performance than the Steam Deck, but for ~$300 more.
