- the plastic glove (and not e.g. hand washing) is the only way to solve this problem
- eating meat is necessary at all, as the OP states, it's not; certainly not in typically accepted volumes, anyway
Plastics have helped a lot in the hygiene aspects of food, as much as some people think (very naively) that food can't make you sick. Especially if you live in a country with warmer temperatures and low infrastructure (hello Climate Change)
If you follow the simple rules, most illness is caused by tainted supply.
The most common infraction I saw was failing to wash hands, followed by not separating meats from other stations. (Work surface, knives, hand washing, etc)
In a busy kitchen, you would be shocked at how much cross contamination occurs every single day, plastic or not. Plastic is not the solution, training and oversight is the essential thing.
For example: delivery of water to locations where the local source is suspected to be contaminated, packaging of meat products from market to consumer, handling of refrigerated or wet products, to name a few
Of course it's not the whole story, a lot of food-borne illnesses have their origin at the producing farm.