Generating haddocks sometimes produces "doc-index.json does not exist"

Edit: this seems fixed in the current 3.8 branch (commit 575b216d8ddc073a26bce5e9139a838e5b344d49). To rule out that the way cabal is built affects it (the buggy version was built by gitlab CI with GHC 8.10.7, the new good version is built by hand using the default `cabal.project` with GHC 9.2.3), I've rebuilt 3.8.RC1 in the same way as the good version and it's still failing.

Edit2: The remaining problems are: we have no test that catches that and we have no clue how it got broken and if the fix is robust enough. Frankly, the only commit I can see that could fix that is the one that bumps process package version. Could a bug in process cause the haddock breakage? I guess it could, given that the haddock exe may be invoked using that machinery.

**Describe the bug**

See #hackage for an extensive testing session by multiple volunteers. It works with cabal 3.6.2, fails with cabal 3.8.0.RC1. Failures have been reproduced with both GHC 9.2.3 and GHC 9.4.1-alpha3. A log from a failed attempt with GHC 9.4.1-alpha3 and head.hackage:

```
cabal v2-haddock --builddir=dir --haddock-for-hackage --enable-doc --haddock-options=--quickjump --allow-newer cabal-install-solver
...
haddock: internal error: /home/mikolaj/.cabal/store/ghc-9.4.0.20220623/directory-1.3.7.1-1b6710e8174fc4ee7baecc57c46ce782434603c3b3b58295a4fcbdc139bbd395/share/doc/html/doc-index.json: openBinaryFile: does not exist (No such file or directory)
```

but at some point (without head.hackage) there was also

```
haddock: internal error: /home/mikolaj/.cabal/store/ghc-9.4.0.20220623/edit-distance-0.2.2.1-a7ea0a2f3ddd1328ec73dacef20a73fff1fc5fe66f9f3edd881ded197bf6a796/share/doc/html/doc-index.json: openBinaryFile: does not exist (No such file or directory)
```

so the problem is not limited to packages shipped with GHC that we override and generate new haddocks for.

We should probably rip out all PRs that touch haddock merged after 3.6.2, in turn, and check if this fixes itself. We should also add a test that catches this problem.

Edit: for the bug to manifest with the `process` package, it needs to be run on the current 3.8 branch (and probably master, too), not on RC1 tag of 3.8 branch.

**System information**
linux