I think this is a bit of self-answering question - if it's D&D, then yes, it's going to allow D&D-style adventurers to walk around, as a default. That's going to be the norm. They might get the evil eye from the town guard or the like, but they're not going to, as the norm, be turned away or the like.
If you want to do something else, you're kind of moving away from default D&D to a more specific setting of your own, which is fair enough, but the "D&D" assumption is certainly that you can.
The problem with demands that people present themselves to elders or the like is that adventurers are likely to be vastly more powerful than the town elders or the like. Immediately attempting to push the adventurers around and force them to conform to your demands, means you're sowing the seeds of hostility.
On the contrary, a more likely scenario is that the elders will politely request the presence of the adventurers, feed them, quite likely give them somewhere to sleep, and so on, in order to set a positive tone from the start, and to help control and direct them into behaving. Trying to bully them or force them to conform is just going to be counter-productive. Unless they're maniacs, you want to start with the "honored guests" approach.
You might not "want" them there, but you don't "want" the King's Tax Collector, or a rowdy band of mercenaries who work for the merchant's guild, or a powerful NPC wizard there, either, but what are you going to do, start shiz with them? Why would you do that? It's perversity itself. These people are, as you've said, more powerful than you. On top of that, most of them are pretty much wildly rich, and will blow serious chunks of change on stuff without even blinking, which could massively benefit your town.
So you have to do what people always do - attempt to have some sort of passive control over them, rather than active control, by directing them positively, because as you've noted, if the town behaves in a hostile way, it's likely to end up in a very bad state, whereas if the adventurers like it, it's likely to end up being able to make $$$ off them.
They'd probably be treated the same way Batman was in the most recent film trilogy - officially, they're vigilantes to be stopped; unofficially they're generally useful to have around, and anyway the local authorities can't really stop them anyway.
Either that, or they'd be treated like rock stars.
Treating rich-ass, superhumanly powerful adventurers as "vigilantes to be stopped" seems like cruising for a bruising to me, so I doubt that'd even be the official line. Historically, whenever someone gets rich and powerful enough, they start getting titles thrown at them in an attempt to buy them off and integrate them into the power-structures of various factions. I think it's very likely that any adventurers who survive for long and are open with their cash are going to have people desperately trying to co-op them, bribe them, and so on, rather than pretending that they're "vigilantes". As for rock stars, yes, that is more likely - even all their deeds are done far from civilization, their gold coins and magic items will quickly make it clear that they're not just making this up.