Enable [[Project:Terminology gadget|Terminology gadget]] by default?

I would say that the gadget expects users to know how to inflect a given word in their own language by themselves. But in any case, both fields in the gadget can use normal wikitext, including templates , so if you make a template that can show inflections of words there, feel free to go ahead and use it. Modules are not available on Translatewiki at the moment, and even if they were, I don't know if there's any way for them to access Wikidata lexemes the same way modules in Wikimedia projects can, so using lexeme data could be tricky. That would be beyond the purview of the gadget, though.

I think just linking forms in the notes is sufficiently helpful that not much else is needed now that I've thought about it.

It is not really a matter of knowing how to inflect words but rather a reminder of which inflections make sense given the context. This is not always obvious from the English source strings, and a number of inflections are exclusively colloquial or exclusively written, or specific to a dialect. Potohari/Mirpuri Punjabi dialects for example allow for future tense while "standard" Punjabi does not, and in the Doabi dialect I am familiar with verbs can have negative conjugations which aren't used in other dialects. So there are various reasons why linking additional context might be helpful, especially for "low resource" languages which don't have as much precedent for software translations.

Agreed, English can be confusing there, especially since the present tense, infinitive and imperative are almost always identical in English. Those ambiguities should be noted in the message documentation, though, since they vary by message (and not by word form), and knowing which one is intended is useful for many/most languages.