If your entire process is broken and toxic, pull requests aren't going to work for you, but that's not a problem with pull requests, that's a problem with your entire process being toxic and broken.
> You make a request... You've been waiting for almost an hour now.
1. Don't design your process so that waiting an hour to merge a pull request is going to be a problem as bad as waiting an hour to eat. Why are you waiting? Move on to the next thing. If it depends on the pending pull request, branch the pending pull request, and work on the new branch--if you have to integrate feedback you can rebase it into the new branch. Occasionally this will cause problems, but they're usually rare.
> Nobody takes a look at and you still need multiple approvals,
2. Choose a number of approvals that is appropriate for your team. If you're writing software where the consequences of a mistake are very high that might be a high number, but in most cases in my career, 1 has been a perfectly reasonable number of approvals. In some cases, 0 is the appropriate number of approvals.
> so you ping Barbara in charge of approvals to get the ball rolling, she says she will take a look after a meeting she is currently attending because she wants to be careful with what she approves.
3. Don't make one team member in charge of approvals--that's almost guaranteed to make that person a bottleneck. Anyone on the team with the knowledge to evaluate the code effectively should be able to approve it. I've been on one team where one guy was in charge of approvals and wasn't a bottleneck, which was because he did nothing but approvals. That worked great for everyone except him--his life was miserable until we scrapped that idea.
> Oh, Tom, her colleague asks you why don't you eat salad and orange juice, it's cheaper and faster. You have a quick call with Tom to explain why you want a coffee and donut.
4. If this sort of exchange is about personal preferences like what you eat or some stylistic nitpicking, then you have to have a team discussion about making sure that we prioritize code review feedback that's actually important and not bikeshedding. But in a lot of cases, Tom is asking because he legitimately doesn't know, and explaining your reasoning to Tom is part of your responsibility to help your teammate keep their understanding of the codebase up-to-date. This isn't wasted time, it's one of the benefits of code review.
> You arrive, by the time you land Barbara approved your breakfast, but now you are in a new country and circumstances changed, so you update your request. The earlier approvals are now invalid.
5. If different pull requests depend on each other or are touching the same areas of the code, ideally do them serially rather than working on them at the same time. If they really need to be worked on at the same time, then you need to communicate earlier in the process to be sure you aren't stepping on each other's toes, and it would make sense for the two people working on the same areas of code to review each other's pull requests. Not doing pull requests wouldn't fix this--this is just a challenge of multiple people changing the same codebase.
> Pull requests assume you cannot trust your colleagues to make reasonable choices
Sometimes reasonable choices are costly mistakes.
Even the best coders make mistakes, and unlike a mistake on what food to eat for breakfast, code mistakes can be very costly.
> and you also don't trust your automated tests to catch anything insane
Automated tests don't catch when your architecture is bad, or when there was a mistake in the specification, or when you've introduced a security vulnerability. Sometimes the mistake caught in code review is in the automated test.
> you have to gatekeep and slow down your team to make sure (?) that the code they commit is good.
Good code can always be better. I, for one, want people looking at my code to make it better. Obviously there are some tradeoffs here, and code review shouldn't be a massive bottleneck in your process. Done well, code reviews speed up your team: the earlier you catch mistakes the quicker they are to fix.