I think having TOO much variety of choice at any particular meal can possibly lead to over-eating in that there are too many choices so that it becomes about trying everything and maybe ending up eating too much. I personally think no-one would be obese if they 'always stopped eating when they were full regardless of the type of food they ate.
I have observed that most of the people I know who are slim naturally usually have one of more of the following characteristics:
By naturally slim his I mean the handful of people I know who don't really seem to think much about their weight and pretty much seem to eat whatever they want without doing that much, if any exercise.
1 - food is really just fuel to them and they often forget to eat for several meals in a row.
2 They are naturally active, fidgety types (though by no means necessarily active in sport and exercise), often quite driven and not what I would describe as laid back at all.
3 They pretty much eat what they want. Suger in tea/coffee, full fat and full sugar stuff whenever they want it, unhealthy pies and chips etc. The key being that they either rarely want this sort of thing or only a couple of mouthfuls is enough to satify them.
4 They rarely if ever comfort eat or use food as a reward to themsleves (this is sort of related to point 3)
Maybe for most of the rest of us having less choice can make us eat more consiously and to be less tempted to eat more than we need and may make it easier for us to eat like this.
The reduction in exercise we do nowadays definitely plays a part for most of us but I think that some people just learn very early to regulate their food intake so they don't overeat. I think this is THE key skill to learn in childhood to prevent obesity.
The problem is that many people will be genetically pre-disposed to overeat a bit so learning that skill is incredibly difficult for some children and easier for others.
In times when most people exercise a lot as part of their lives and food is not so widely available then it is much less of a problem for those who find this skill difficult and haven't learned it but in the times we have now obesity is an almost inevitable outcome for them.
I think though the choice should refer to the number and variety of food on offer at any particular time not the choice about whether to eat it or not.