> I thought there might be a chance that the bare infinitive in e.g. the conditional mood is derived from an older true mood...
CGEL says this:
Given that the three constructions in [24] always select identical verb-forms, it is inappropriate to take imperative, subjunctive, and infinitival as inflectional categories. That, however, is what the traditional grammar does, again retaining distinctions that were valid at an earlier stage of the language but have since been lost
[I'm taking a position that disagrees with this one, but it does address the use of mood in earlier stages of English.]
I feel that that may not directly address your specific question, but it's hard to know what that question is, since English conditional clauses do not use the form you identify as 'bare infinitive'.
> ...an older true mood (in the linguistic sense, i.e. a verb form that is sufficient to signify the mode)
This doesn't make sense. A verb form is never sufficient to signify the semantic mode of a sentence. Nobody ever argues that Latin didn't have inflectional mood, but good luck identifying why a verb appears in the subjunctive if you can't see the rest of the sentence around it.
(There is a whole traditional taxonomy of different Latin subjunctives; the most common cases are conditional clauses, which use subjunctive mood to indicate counterfactuality, commands ("jussive subjunctive"), and wishes ("optative subjunctive"). Another case is "the verb is part of an indirect question". [Do indirect statements use subjunctive verbs? Nooooooooo...])
So, for English, we have a distinction in semantic modality that obligates us to use an exotically-conjugated verb. Why is this not an example of grammatical mood?