Some of this makes quite bemusing reading as so many ingredients are loved by some and loathed by others, but that's just personal preferences. Non veggie dishes are no different. Duck, venison, and black pudding are probably the meat equivalent of goat and blue cheeses
This is it in a nutshell. No one would battery and eyelid if meat eaters didn't like those foods - and most of the time it wouldn't even come up, as they'd choose one of the dozens of other dishes on offer.
I think the main thing you should think about for your pub/restaurant is choice. It doesn't matter if ten people on here have said they hate mushrooms; if they could choose one of two or three other options it wouldn't be an issue. However, I would personally avoid goat's cheese tarts or similar, as it suggests an imagination bypass.
New careful about attempting cuisine you don't make for meat eaters - eg if you don't do meat or seafood pasta, don't make veggie pasta your only option. It wouldn't reassure me that your chef was any good at Italian food.
Mushrooms are ubiquitous because people assume veggies are all on the verge of death from lack of protein. Peppers get chucked in everything to 'give it a bit of colour' (the irony being that most meat is brown and most fish grey...) Try underused vegetables - cooked beetroot is getting more popular, celeriac is an unusual alternative to butternut squash, and red cabbage is a tasty splash of colour.
Whatever you do, pay attention to your sides. I remember going to a well-regarded English restaurant years back with family and seeing the most amazing vegetable accompaniments coming out with everyone's meals. Then my lasagne arrived with a bit of limp salad. I despaired.