You wouldn't think Starfleet would want to leave a fully armed ship lying around for just anyone to find.
Starfleet might be too short on resources to do anything about a ship lost so far out that the evacuees would "drift for weeks" in their shuttlecraft. OTOH, it may be they did go and have a look at the system, despite knowing an enemy stalks there - and found nothing, as the Ferengi had already towed away the hulk.
Canonically we have no idea, "The Buried Age" by our very own Christopher covers this very nicely. in the book the ship was overly contaminated by radiation when the crew evacuated and the auto-destruct system failed.
The problem with that is that Picard himself doesn't seem to think the ship would have been destroyed - that is, it appears he never made any attempt to destroy her. He's happy with her "finding her own way", in a log entry apparently made just
before he left the ship. Nor is Picard sufficiently surprised that the ship is still intact in "The Battle".
So a botched scuttling attempt doesn't really fit the picture. The idea that the ship would have been difficult to scuttle is all right and well - we know autodestruct charges can malfunction. But apparently in such cases Starfleet does not require the skipper to go out of his way to destroy the ship by some other means; leaving her lying about is acceptable. (In contrast, no dialogue suggests scuttling would be a required maneuver in any context, and our heroes typically only use scuttling as a blackmail tool or a way of taking a few enemies with them.)
As for the 7100 registry? Aside from what's been already stated about reusing the decals from the AMT kit, there's another way to look at it in-universe: The model could have just been a representation of his ship class.
Makes sense. OTOH, in the age of effortless replication of trinkets, it could also be that a generic model was replicated and then a very special number was painted on it before the item was handed over to Picard as a gift. Perhaps 7100 was the transporter number of an attractive lady from Picard's past, or his score in the Academy Septathlon, or the sum he owed Walker Keel?
That excuse has been used so many times on screen it's ridiculous.
Let's remember that basically none of our heroes are suicidal. Auto-destruct is not a means of committing suicide, it's a means of scuttling the ship without harm to the crew - hence the countdown and the warnings and all. If the countdown timer doesn't work, there are a zillion ways of destroying the ship - but few or none to do it so that the crew survives.
Timo Saloniemi