Nintendont can support any video mode natively supported by Wii mode, including 240 and 288p. But just because it can it doesn't mean it currently does. No guarantees on games working, but games that linedouble to the normal standard definition interlaced modes should.
I recall something about 576p support being purely restricted to Gamecube mode, sadly. Not sure if it's true. It's unfortunate if it is, because some PAL games are restricted to 50hz, and don't play nice with being forced into an NTSC aspect ratio and refresh rate. (Even patching will usually make them run too fast.)
At least it's not like PS2 where you have to choose between RGB and it's sync-on-green nonsense and YPbPr and the annoying combination of modern TVs not liking 240/288p over YPbPr, some PS1 games mode switching, and certain PS2 games having interlaced framebuffers (So NO progressive scan support even via forcing).
As much as I've grown from experience to really hate CRTs, they are at least easier to manage most of the time.