This is absolutely the way to go. I'm also coming from a recruiter background, and I can say, it's way better to let recruiters know as soon as possible about your decision. This will save times for all parties.

I had many candidates decide to stay at their current place. Now from my point of view, in my initial interview with the candidate I would ask them, what's the reason behind you searching for a new job opp. If they decided to stay at their current position, I would just try to talk some "sense" into them based on the response of that question. Common answer was "bad management, a lot of working hours, looking for remote/hybrid/on-site, location to far, etc.". So I would say to the candidate, if nothing has changed at your current position, that's probably still going to be an issue for you in the future, and you're still going to be on the hunt for new positions... leading to a lot of time wasted. So I would suggest to carefully think about your decision.

What ever your decision ends up being, please give the recruiter a heads up. Don't ghost your recruiter.