OP - I have had my horses barefoot, I have had them shod. It depends on the job they are doing, the quality of the hoof, and the comfort level of the horse. It is not evil to put shoes on your horse; if your horse is tender, doesn’t want to walk on rocky ground, etc. without shoes then it is not (currently) a good candidate for being barefoot.
Shoes vs. hoof boots – if your horse is comfortable except for specific circumstances, such as being ridden on rocky trails, then hoof boots can be very helpful. I’ve used Cavallo and Easy Boot hoof boots, depending on the shape of the horse’s hooves. For awhile I even hunted in hoof boots.
Wedge pads – these are a double edged sword. If your horse needs to have its angles corrected, they can help. But I’ve also seen farriers adding wedge pads for the wrong reasons and then it becomes problematic to take your horse out of wedge pads.
Soreness after trim – this is not good. Your horse should not be sore if it is done correctly. Even when I took my horses from shoes to barefoot, they walked off sound. I agree that you need another farrier or at least another opinion.
Farrier vs. vet - Not all vets are good at evaluating shoeing issues, but if the underlying cause is medical (not just removing too much sole or bad angles), they should be consulted. Unfortunately, I’ve also come across some terrible farriers who left my horses with toes that were too long and feed that weren’t balanced properly. It’s very hard when you are a new owner to assess something as technical as shoeing.
It sounds like you are being “bullied” a bit by your BO and the farrier to let them do business as usual. There is nothing wrong with asking questions. If your horse needs wedge pads, asking them why. If they can’t give you an explanation, bring in someone else and ask the same question.
Good luck!