I do know how easy it is to get in the zone as a developer, and subsequently feel frustrated that a useless meeting just popped up in the middle of your flow, but that's not a good reason to write off 1on1s entirely. Worst case your flow will be interrupted once a week. Two if your team has a weekly retrospective as well. If two interruptions per week is significantly damaging your productivity, something is wrong.
Personally, I am not a fan of meetings in general, and I do consider many of them to be useless. But 1on1s and retrospectives are the two that I mandate in the teams that I lead - one is for the health of my reports, and the other is for the health of the team. All other meetings can and should be optional. If someone is feeling snowed under by all the other meetings, then it's exactly during the 1on1 or retro when those feelings can be shared. If you don't have those meetings, then you're relying on everyone on the team to always speak up whenever they need help, and while that might work for certain, very assertive and/or self-centered employees, it doesn't work for everyone. For that reason I think it's important to schedule these times in deliberately to give people the opportunity to express themselves.
When there isn't a culture of open communication - which is established through regular meetings of this sort - many employees find it difficult to be honest about their challenges and difficulties (on the negative side) or goals and aspirations (on the positive side). In either case I believe it improves productivity and retention to give employees dedicated space to talk about that stuff.