For some vegetarians and vegans, they are greatly troubled if they consume even any animal by-product. There was a point when I felt this way; I stopped after a while, and told people "Don't worry, I'm not religious about it."
I think there can be digestive concerns; eating a non-meat thing that was soaked in beef-grease may cause digestive trouble for someone who hasn't had beef in a very long time. But I've also heard of former vegetarians who had no trouble when they started eating meat again. Personally, I've suspected if some minor digestive trouble after various meals was caused by stray meat by-products making their way into my food, but I can't be sure what caused it.
Wouldn’t those same people want to avoid a place like Burger King in general? They’d be supporting the business of one of the largest consumers of beef in the world.
There's value in supporting ethical businesses. There's also value in encouraging less-ethical businesses to be more ethical.
I can't speak to the situation in the US, but here in the UK, veggie burgers in fast food restaurants represented a fairly significant tipping point. In the late 80s and early 90s, it could be genuinely inconvenient to be vegetarian. Asking "do you have a vegetarian option?" in a restaurant often resulted in a blank look. If you wanted to buy meat substitutes, your only option was a health food shop.
By the 00s, being a vegetarian had become a complete non-issue. Every restaurant had at least one reasonable vegetarian option, every supermarket had a decent selection of meat substitutes and vegetarian ready-meals, every product that was suitable for vegetarians was labelled as such.
For would-be vegetarians, that ubiquity removed a significant barrier. The sheer convenience of vegetarianism induced a lot of people to give it a try and took away an excuse for not trying. Perhaps equally significantly, it normalised vegetarianism - when McDonalds and Burger King offer a vegetarian option, you can no longer argue that it's some weird hippie fad. The same thing is now happening for veganism.
I’m not clear that this makes Burger King any more ethical, or less unethical. You wouldn’t praise a prolific serial killer for letting the odd victim go. Besides, fast food has a lot to answer for besides animal welfare.
On the other hand, by purchasing vegan options at a fast food chain, they would be increasing the market signal for vegan food. Getting a massive chain to reorient its supply logistics so that 0.2% of inventory is vegan instead of 0.1% could have a huge impact on overall animal welfare and environmental welfare.
Some, yes, but not all. Everyone has to negotiate their personal principles versus practical and social constraints on a daily basis; this is just an instance of that phenomenon, and not everyone will do it in the same way.
I think there can be digestive concerns; eating a non-meat thing that was soaked in beef-grease may cause digestive trouble for someone who hasn't had beef in a very long time. But I've also heard of former vegetarians who had no trouble when they started eating meat again. Personally, I've suspected if some minor digestive trouble after various meals was caused by stray meat by-products making their way into my food, but I can't be sure what caused it.