Your examples are examples of the use of 'better' as a straightforward adverb. Though like most comparative adverbs, it would usually follow the object, rather than preceding the verb.
It better shows the color of your eyes would usually be
It shows the color of your eyes better [than the other options] in America. Your usage is not outside some idiom, it is just so close to a common idiom that people avoid it.
OTOH two of your examples are off-center from another idiom: 'sense', 'help' and other pure virtues have a quantity, and not a quality.
Something does not 'make sense well', so it cannot 'better' do so. It 'makes a lot of sense'. Similarly something 'helps a lot' and does not 'help well'. All sense is good, and so it all help, there are only degrees. (There is an exception where one mechanism helps or makes sense of the same thing in a way more fitting to some situations than others, but it is not relevant to normal speech.)
Matching has a quality, as do most more-active verbs. So one thing can better match another than other options. Showing, explaining, etc. can be done well or poorly. So 'better + verb' makes sense with the right kinds of verb.