Legally speaking, businesses and people are free to discriminate against anyone that isn't part of a protected class[1], but many fail to understand that and assume it's some sort of slippery slope.
I argue that the progressives you mention have no issues with business discriminating against certain groups, provided they don't do it against protected classes. Objectionable political content, like inciting vololence or discrimination against certain groups based on their race has already been ruled by society as not ok. Hence why they're a protected class.
Certain groups such as LGBT are in need of further clarification as a protected class by the courts, as there are conflicting interpretations depending on jurisdiction. This is why the Colorado baker case is so important [2].
[1] https://en.m.wikipedia.org/wiki/Protected_class
[2] https://mobile.nytimes.com/2017/06/26/us/politics/supreme-co...