I wouldn't try it anyway—I've never understood the "fake meat" thing, it's just not in my brain I guess—but yes: grilling it on the same grill as beef, chicken etc. is a dealbreaker.
Basically, for the same reasons gelatin, lard, chicken broth, bone broth, fish sauce, bacon bits, etc., are dealbreakers. I do like that they let people request they be grilled on a non-meat grill. I suspect the default is just for meat-eaters who want to try something airquote-healthier-airquote.
I'd say Burger King can expect a backlash, but in my experience vegetarians' expectations of fast-food places like that are very, very low: "we" don't expect to be able to eat anything on offer.
as LouNetizen says, this entire movement seems more geared to 'flexitarians', who won't care. Most of the vegetarians, and nearly all of the vegans I know are deeply suspicious of fast food joints; often for good reason
I've had places swear to me they didn't use animal-based oils, only to discover they'd lied. While cooking something on the same grill used for meat probably won't kill me, it sometimes spoils the taste for me to the point I haven't finished a meal
and then, my favorite story: ordering a quesadilla in a restaurant in Tennessee after explaining I was a vegetarian. When it arrived with ham in it, I was told, straight-faced, "That's not meat... that's ham"\
Yeah, even though I'm not actually either vegetarian or vegan (I'm pescatarian), I feel the same. Although not all veg*ans are the identical, I think it's safe to say that avoiding food cooked on the same surface as meat is the norm, and those who don't mind eating that are the exception to the rule.