I worked for an agency (web development) and had similar experience to you. You mentioned many things, but one of them you are describing I think is a common and possibly the biggest/hardest issue to be solved in this business model, which is workload. We either had too much or too little to do.
So when you have too much on your plate, obviously you need another person to delegate some tasks to. If you employ someone, the workload may diminish after six months, so what does that person do: train, learn, work on an internal project? New employees also need training, it is hard to find the perfect person that can come and immediately start closing tasks one by one without some help/onboarding. So when a new person comes, things do not go faster at first, they actually slow down, because time needs to be spent on the new person and he needs time to learn the projects and adjust.
Other things you mentioned are focus and priority.
How I managed to somewhat focus under high stress and try to not get completely burnt out: - I was working on the tasks one by one and not looking at the pending and backlog tasks. Just work on the current one and not care about the rest. Finish the current, take the next, and so on.
- Trying some time management techniques like pomodoro, especially when working from home. At least it gives some structure to time.
- Not looking at emails or group chats on slack for 2 hour chunks.
- Unless there is something really urgent, stick to the working hours in the contract.
Priority is another big one. There were many/most of the time(s) where I had more than 5 things pending. That is not an issue for me, as long as I know what to work on currently, what is next and so on. This depends a lot on a good manager, because it can add/reduce stress in already stressful environment. With the fist manager I had, 'urgent' meant, stop everything you are doing, (you can also skip some processes like reviews, deployments) but get this fixed. Usually some site was down or some other big issue and it rarely happened. But, with the next manager, there where 1-2 things urgent almost every day. Which added a lot to the defocus and stress.
I did burn out after 4.5 years though, so did the people from my team. We were 12 people when I went in and I was actually one of the last man standing.
I think, these are common issues that you are describing, for an agency and programming in general. If you fix them, you have created the 'perfect' agency and became the 'perfect' programmer. Which is probably really hard, but a good thing to strive for.
Good luck on your journey!