On the other hand what could be very useful is some kind of marketing/business development person to help with finding clients.
But as a freelancer myself, the problem was always just earning enough money - so I never hired a coach or PM. I guess theoretically they can pay for themselves by getting you to finish more billable projects. But you have to be really objective and honest with yourself about whether that's actually happening.
I think you are correct in your thinking. Freelancers should hire a PM.
Given that I am serious about my freelancing and want to build a real business around it going forward, I have to be serious about running my business. That includes things like setting priorities, client comms, and business development (=sales).
If I can't or won't be serious about these things, I won't have a business. And, eventually, my practice will look more like employment with a different paperwork (and no benefits).
Coaches can be very effective, though. I've had a session with a retired exec thanks to SCORE, and her advice was valuable. Getting actionable advice about running your business can bring measurable financial benefits, e.g., in your close rate, or your pricing.
Peer groups can have a similar benefit. I am a member of Jonathan Stark's group coaching Slack, and I learn something new every day.
I think it's important to address one's personal weaknesses that are related to one's business practice, and work on them rather than delegate. I am happy to outsource my accounting but I wouldn't hire a PM to talk to my clients on my behalf. Rather, I'll work on my communication skills so that I don't embarrass myself in front of a client :D
If you're a PM looking for a role as a freelancer, I'd consider taking the consultant route. If you're a PM with a deep experience either in a particular industry (e.g. finance) or a particular technology (e.g. APIs), you can easily pivot to a consultant to those types of companies, helping their PMs and teams figure out how to compete.
But even when I was a full time freelance developer focused on producing code, though I would not question the value of having a PM, I specifically freelanced so I could manage myself.
Note: do _not_ share this with your client! Client channel should be separate and are dependent on on the client's preferences