As you point out, going by the typical doctrine of extractive institutions from Why Nations Fail (Acemoglu), we have this basic framework:
Disenfranchisement of private sector leads to capital flight / brain
If long-lasted, this leads to slower economic growth and potentially
a failed state
However, I think we may want more granularity here. A ...
A very good paper that I had in my MA curriculum is Jaychandran and Pande, AER 2017: https://www.aeaweb.org/articles?id=10.1257/aer.20151282 (I believe this paper has open access, maybe on one of the authors' websites).
Working through this paper should help you understand DID quite thoroughly. The paper also employs double and even triple DID.