Are Mathematica commands always supposed to check that Method -> name is valid?

The other answer is completely correct (see my comment), but let me add some context. For algorthmic functions, it is generally the case that we validate the method options and issue messages/fail (sometimes a fallback is used). However, in "container" type functions, like Graphics/Graphics3D, we often don't validate them. This is because they may be used by some layers and not others (for example, in Graphics, the value of Method might only affect front end rendering, and it is the front end which must understand it). If they were validated, then every layer would have to know all the valid values, even if they were completely irrelevant.

I would expect a fully implemented/documented function to check the validity of the specified Method; however, since AsymptoticDSolveValue is marked as [EXPERIMENTAL] I would not assume that it is fully implemented/documented.