As for the number of analyses you can run, that depends on what you mean. You're right that differential privacy won't allow you to set up a database of _confidential data_ that can be arbitrarily queried infinitely many times with any meaningful privacy guarantee, but this is in no way unique to differential privacy.
What you can do with differential privacy is release noisy statistics once and let researchers use those statistics for arbitrarily many analyses. This is what the 2020 US Census is doing, for example.
> For official statistics and scientific research this is often not an acceptable tradeoff
is that differential privacy is the best known method for rigorously accounting for privacy risks. It's possible to argue that differential privacy is too strong (and plenty of people have), but to the best of my knowledge, systems that say "you don't need DP - we'll answer lots of database queries without DP and still prevent deanonymization" usually end up getting broken. A good example of this is the (repeated) breaking of Diffix [1], a system that attempts to provide privacy without using differential privacy.
So differential privacy is, I think, a good starting point if privacy is critical to your application. It does not offer much guidance for when you should decide privacy is critical, or when the utility of an application outweighs the need for privacy.
For example, many social science researchers have criticized the US Census for using differential privacy in the 2020 census. It's consistent to say "it's way more important to have accurate counts for all of the decisions made using census data -- let's not try too hard to be private". It's also consistent to say "privacy is important, so we should use a rigorous notion like differential privacy". It's not consistent to say "private is important, but let's just use some heuristics and hope for the best", which is what the census had largely been doing until 2020.