I think they're mixing up cause and correlation. My theory is flying animals evolved shorter guts to save weight, and it's the shorter guts that cause a lack in biome diversity. The weight of the gut bacteria is irrelevant, it's just a side-effect of gut reduction which probably constitutes a larger % of bodyweight.
The causal link suggested was because flying animals would need to be light, their digestive system evolved to be shorter than similar land based animals and also save the weight of the microbiome passengers(or it could be with shorter digestive time, no bacteria evolved to be symbiotic quick enough to be beneficial), so there is nothing like the symbiosis between microbiome and host animal that we humans and all other animals with backbones possess (phylosymbiosis). I'm curious how much they sampled non flying birds - it seems like a reasonable test for their hypothesis. There was some bacteria in flying animals guts, just not consistent in species like it is in animals that need phylosymbiosis or as much as non flying.
Gut bacteria have beneficial effects, such as the ability to break down foods which would otherwise remain indigestible. If birds and bats have reduced gut microbiomes then it suggests they have a reduced ability to digest certain foodstuffs.
Also: the weight loss referred to is the reduction in gut bacteria mass.
Sure, it's a positive effect on your ability to fly. But if you do anything other than flying, it will affect those things too. It's terrible for your ability to recover from injuries.
