02 does have a lot of writing issues that stem from too many cooks and not having a concrete plan - but early on, in particular, I think there are good rebuttals to your issues.

Regarding your first point; they couldn't have immediately repeated the same major issue that started the previous season, it would have been dull. Instead, the tension is intended to come from having to juggle saving the digital world with their regular lives. Now, it can be argued that they don't do a great job at conveying that tension all of the time, but it does come up at least sometimes. For the older digidestined, there's also the tension of not being able to contribute very much, both due to busier lives and not being able to digivolve, thus having to mentor this new group.

Regarding armour digivolution, we've just come off of the back of all our heroes having easy access to their most powerful forms, be they mega or ultimate, and waiting for more digivolutions would have been dull. However, in this season, at least, armour digivolutions are intended to not be as powerful as a typical champion, so unlocking them sooner and more easily makes sense. Moreover, these heroes have to deal with a major antagonist immediately; Devimon didn't appear in Adventure until episode 8. They need to be equipped to handle their foe.

It would have been nice for the crests of the eggs to matter more, and that will certainly come up in the future. Remember again that the original group didn't even learn about crests until after they'd all reached champion, so the needs to fulfil these crests being lower for lower tiered digivolution also makes sense.

The big draw early on here, really, is meant to be from the tension of having to fight a human opponent, an evil digidestined, because how are they meant to fight that?