That's not a bad idea at all. I second that idea.
To expand on the idea, I find that the actual "workout" part of climbing is not super effective... I mean, walking/ hiking is a workout too but by itself it's not going to bring about many physiological changes without a lot of additional targeted work. For context, I don't climb "hard"... I lead trad around 5.9+, sport around 11a/b, and was grunting up some WI4 this weekend as a newbie ice climber in Ouray-- I don't boulder outside of a gym if I can avoid it.
I think most people can climb 5.10 outside without changing a lot of fitness... the skills and techniques are so much more important than strength, assuming a basic level of fitness.
What I have found is that I'm a lot more likely to do actual working out when I have a project I'm trying to get strong enough to do.
Additionally, climbing keeps me doing approaches with a pack, and there is fitness that comes with that practice. Or I need to get up hella early, so it often prevents me from drinking. Or I have a reason to fast and keep my weight as low as possible without feeling like I'm jut being neurotic and picky about what I eat.
That is all to say, I've gotten a lot of stoke for working out from climbing, but the climbing itself hasn't been specifically great workout wise.
So maybe this is just a quibble, and not a disagreement; I certainly like to recommend that people get out and climb and it sure has gotten me around the issue of not wanting to exercise, even if I don't find it to be a great workout in itself.
But that quibble does change the character of the recommendation, I think: in general, I think that any hobby which benefits from high levels of fitness is a good thing that makes working out easier. Mountain biking, back country skiing, mountaineering, canyoneering... all of those are great and make getting into the gym feel like it has a much higher pay-off.