On the subject of Chomsky and universal grammar, one of the parameters that is used to classify languages by their basic features, along with the order of subject, object and verb, the existence or otherwise of a nominative/accusative or ergative/absolutive, whether it is synthetic/agglutinating/inflected/analytic etc, is whether it is or is not pro-drop. This inelegant term refers to the possibility of not using an explicit subject with the verb, and sometimes includes ellipsis of the object as well. Neither Latin, nor Greek nor Sanskrit required a subject where it could be identified from the context, the deixis or the termination. Modern Spanish and Italian do the same thing, but English and French do require it in almost all circumstances.
Not only is the term inelegant but I wonder how much significant information it really gives about a language. The use of pronouns seems to be required, broadly speaking, only when the subject would otherwise be unclear, either because the verb morphology does not discriminate sufficiently, or, especially, in the third person, where the range of possible subjects is very large and identification must take place somehow. It should be borne in mind, however, that this can be done non-verbal deixis, and in any case need not be repeated once the subject is known. In languages such as English, where it is very common to have to identify a subject explicitly, the convention has arisen that it always be done, but this does not seem a normal feature of languages.
A more important point than exactly how a language chooses to identify a subject is that it always does. I cannot state that it is a general property of language, or a feature of universal grammar, but I would hazard a guess that it is so. It is certainly present in all the languages of which I have enough knowledge to judge (which, admittedly, include only one non-IE language, Basque).
The verb gives four essential pieces of information. It tells us what action was performed (the main verb), who did it (the subject), when it occurred with respect to the speaker’s present (the tense) and its relation to a second time reference (the aspect). This information may be encoded in different ways, in a single root with morphological variations for each combination, in a variety of different roots, with one or more words for each point, as in the isolating languages, or some combination of these options. English almost invariably uses at least three different words, with the aspect encoded between the choice of tense indicator and morphology on the main verb. Many languages encode the subject into the morphology of the main verb, and so often need no separate marker for it. But it is there, and can be identified.
The first time reference is always the speaker’s present. The second is redefined constantly by the speaker, using verbs in the ‘simple’ tenses, or by an adverbial phrase or clause, or by an external reference, explicit or implicit. The aspect, broadly speaking may be simple, which indicates that the action of the verb coincides with (and as mentioned above, actually defines) the second time reference; it may be continuous, in that the action takes place across the time reference; it may be perfect, in that the action takes place before the time reference. Not all languages have forms for all these, and some doubtless have more, but they are the basic ones and the differences can always be expressed somehow. Subjunctives, optatives, conditionals and so on are deliberately outside the real time system, because that it is what they are intended to express.
In case this is all far too tedious (or just wrong), I offer you the picture of subject, action, tense and aspect.