No
Or at least, we could use the materials, but it wouldn't be enough:
The total mass of the inner asteroid belt is only equivalent to about 0.7% of Mercury's mass, so according to the material requirements from the video, you would not be able to complete the construction project of a Dyson swarm using only materials/mass from the inner asteroid belt.
As an additional note, you wrote:
If one wanted to mine Mercury in the future, I bet many people would be upset about it
While there would probably be some people upset about this, I don't think it would be "many" in a percent-of-population or even percent-of-experts sense. Mercury, as far as we can tell, is mostly just a sun-blasted hunk of material, and unlike Venus or Mars, much less interesting from a scientific perspective (basically no life prospects). It has had comparatively few missions to it (only three, one currently in progress).
Also, while hard to quantify, I don't think that Mercury has the necessary amount of "social capital" associated with it to prevent disassembly if there were a good reason to do so (unlike, for example, Mars or Luna which have many cultural and social bonds to protect them from planet-scale mining operations).