I’d classify Thexder more as a run-and-gun platformer. On wikipedia it’s even claimed (so it must be true ~_^):
> Thexder is considered an important breakthrough title for the run-and-gun shooter game genre, paving the way for titles such as Contra and Metal Slug.
Source: https://en.wikipedia.org/wiki/Thexder
It has a rather unique “transformer” mechanic, allowing for the bot to be changed into a jet at any time.
If the dos version has the same (lack of a) save-game system as the msx version, it is also tough as nails.