Should overweight people climb? If they want to, then yes of course they should.
I don't have any hard fact to back it up, but I do know a few climbers that are slightly overweight and have participated in guiding climbing days where some participants where seriously overweight (in some cases due to obvious medical reasons, in other cases I don't know why).
As climbing is vertical, there is an obvious disadvantage of the additional weight. This even applies to friends who do cycling and climbing: They usually complain that they need to lift those huge thighs up the wall. Also looking at pros like Adam Ondra (especially in his younger years) and at kids around 8-12 years, the incredible strength to weight ratio is apparent.
Now that all sounds like I am arguing against climbing when overweight - far from it. You need to adhere to the same principles any beginner or anyone on re-entry needs to: Listen to your body, don't pressure yourself or let anyone pressure you into doing anything you don't feel comfortable with. This is especially hard if you climbed a certain grade some years ago: You won't reach it immediately. Go at an easy level such that you enjoy climbing and start raising the level in a way you feel comfortable with or stay at the level you like, nothing wrong with that.
No matter if you are perfectly fine with your weight and just want to climb, or if you want to climb and lose some weight, climbing is a great activity to do it - don't let yourself be talked out of it by obviously badly informed peers.
Addendum (not directly relevant to the question):
As other answers and comments tend to go in a different direction, let me give a motivation for this answer, skip it if you don't care:
I know the OP does already do high-altitude mountaineering, so I considered this to be about rock-climbing in the narrower sense, not climbing as in every activity involving non-horizontal motion in the broader sense. Also the question is centered on social interactions, being negative about climbing while being overweight - that's what my answer addresses. The question does not mention any technical issues about climbing. There are technical/security related aspects to be considered, but there is nothing preventing you from climbing. This is nothing against other answer, it just provides my reasoning why I don't include such points, as brought up in comments, in this answer.