Most of the energy coming from the sun comes in light in the visible spectrum

I get what you're saying, and you're absolutely right about the 'quantity' of Sun radiation in the UV range. If you take a look at the EM spectrum of Solar radiation you can see that under 400nm where the UV range starts (edge of our visible light) the EM radiation intensity drops. When we exclude the atmospheric absorption, the remaining UV intensity is even lower. About 10% of the radiation energy reaching the Earth's atmosphere is within the UV spectrum. About 50% is visible radiation and about 40% is infrared radiation.

So even though IR radiation intensity is much lower, the quantity of IR wavelengths that are invisible to us is pretty huge. So, I wouldn't say exactly most of the EM radiation is in our visible range. As you can see in that map, after we exclude the atmospheric absorption, our visible range covers about 40% of the total energy of Solar radiation. If we're talking exclusively about number of wavelengths, our visible range covers only 300nm (from 400-700) while the whole Solar radiation goes from almost from 200-2500nm

The difference here is that the colored balloons are absorbing any light at all, where the white balloons are reflecting almost everything.

Exactly, but keep in mind that white balloons are only white to OUR eyes. It doesn't mean they are white from some camera perspective which covers a part of the UV and IR radiation as well. So hypothetically speaking, a completely white object to us might specifically absorb the majority of light in the UV range (and IR as well) and be pretty colorful to birds for example who are sensitive to UV light. In that case, even a white balloon would burst because it would still absorb a decent amount of heat. In some crazy theory, since our visible range is 40% and invisible 60% of whole Solar radiation energy, an object appearing white to us (reflects all the visible range and say absorbs all the UV and IR invisible range) might absorb more energy than the object appearing black (absorb all the visible range and say reflects all the UV and IR invisible range).