But with purported past tense 'could' we are unable to state the opinion that he was or wasn't there.
*He could be there. // *He couldn't be there.
Same for "can". Without the perfect, it doesn't point to a finished event.
*He can be there. // *He can't be there.
The two modals, can & could don't carry tense into the sentences, they carry modal nuance. The tense is shown by "have + PP".
I think we're getting caught up in the imprecise terminology. Past tense is a morphological distinction: that is, it's just the form of the word. We call it the "past" tense, but that's just an inaccurate label.
We often do use the "past" tense to describe actual past events, but we can do more:
If I had a hammer...
is a past tense used to indicate not a past event, but an event (or, in this case, actually a state of affairs) we consider to be hypothetical (I don't have a hammer).
Modal verbs are no different in this respect. Thus we can utter the sentence "I couldn't speak Spanish last year, but now I can", and this manifestly contrasts past inability with present ability.
But we can do more:
I can speak German. I could probably speak Russian if I tried hard enough.
The past tense form (which is an inaccurate label for a morphological distinction) is used here to indicate a hypothetical ability, which will remain hypothetical at least until the condition is fulfilled.
But we can do even more. In the same way that the verb "to have", in addition to its basic duty as a verb meaning "to possess" can also be used as an auxiliary to indicate aspect, so too can the verb "can" perform an extra duty and indicate modality.
Modality has to do with possibility, certainty, obligation and so forth. When "can" is used in this way, the tense forms are used to indicate not time, but degree of certainty.
In that sense, it essentially becomes "timeless".
But not actually "tenseless", because tense is, in its strictest sense, a morphological distinction. Thus Random House has for its primary definition: "a category of verbal inflection that serves chiefly to specify the time of the action or state expressed by the verb" -- "chiefly" here clearly implying "not merely".
Confusingly, we not only label the tenses (misleadingly) "past" and "present", but to make life easier on our students, we usually use "tense" in its much wider, looser meaning of "the time, as past, present, or future, expressed by such a category" (Random House's third definition).