Of course we still have those things like aspect etc., but as you also say, we can't express them briefly anymore. In English, you need three words to express 'I have done', in Russian you need one and in Latin too (I don't know about Polish, maybe a little more than Russian).
You are right that for Slaves it would become a problem to communicate without cases. Apart from the odd preposition; what's the difference in meaning only between 'a sem c nim' or 'a sem s on'? Granted, it sounds weird, but in terms of meaning, you'd probably understand. If on the other hand, you were to leave out a dative case in a long long sentence, the direct and indirect objects could get swapped around in hte mind of your listener. Chaos abound!
German on the other hand has fixed word order. It's daughter language Dutch has got no cases anymore and we manage fine. The only real remnant of the old case system is our personal pronouns which are structured in two tiers like in English (although some linguists insist on a separate form for the third person plural dative, but that's nonsense). Germans could manage perfectly well without cases, I think and it's going that way. In spoken language, they don't use the genitive case anymore, but use 'von + dat.' instead.
But, may I ask, what do Poles do if they have to express a past perfective idea?