I grew up next to a farm. I am fully aware of where food comes from.
Perhaps, but so did many other people and they didn't make the connection between their meals and dismembered animal parts with lots of blood (or non-human animal screams, obvious pain/torture they are under - despite of how 'nice' or 'humane' the farmers make out the process to be on the surface) until they actually saw it for themselves or were asked to take the animals' life.
Yes, some people are raised to be hunters, but in order to do so, they also need to desensitize a person to killing for one thing and also allow you to perceive non-human animals (and make no mistake, humans are animals as well) as 'lesser' than humans (which realistically, they aren't) - and just for reference, people who DO work in slaughterhouses actually end up with a variety of disorders (such as PTSD and a bit less well know PITS).
There might be more to this line of thinking which you might not have included in your thought process (but admittedly, I do not personally know you, therefore, I cannot presume to know what you think or how you feel or what you may or may not know).
That it is waste products, the smell, how ill it can make people if not handled appropriately, among other reasons.
By that reasoning, shouldn't the concept of eating regular food disgust you as well?
Majority of food we eat is fertilized by non-human animal waste (which is degraded by bacteria, worms, etc. in the soil and then used to 'feed' the plants - which end up in your mouth).
Heck, how many humans were buried throughout history only to become worm food and something for the plants to thrive on (which we eventually end up eating in one way or another).
As for how ill it can make people if not handled appropriately (or how bad it smells)... sure.. there's a possibility for that... much like there's a possibility of food poisoning from ingesting salmonella which contaminated the plants due to animal feces that came into contact with it (practically very little to none if you handle the food properly and wash the produce in question - oh btw, animal products turned into food for humans are far more frequently contaminated and recalled compared to plant foods).
Recycling is essentially a process that not only removes contaminants, it re-arranges the matter base structure to the point where that matter becomes a completely different form - essentially its a far cleaner and more controlled process than I'd say nature even (especially if automated).
I really don't see the rationale behind the feeling of disgust because the things you describe about recycling (at least from your perspective) apply to how we grow plants.
And by that line of thinking, if recycling in the 21st century can be efficient, sustainable and produce all kinds of things with better quality without there even being a hint of what it originated from (which it can and does), you can bet that Star Trek 32nd century atomic rearrangement of matter would be far more efficient, cleaner, safer and probably contain more nutrients than what a 'real apple' can provide (in fact... it can probably also be made to be sweeter and nicer tasting than the real thing - so this 'concept' that replicated food is somehow inferior is actually bogus and further reinforces the premise the writers don't understand certain things or are just plain lazy).