First, you're implying that all opinions are equally valid. That is... I'll be polite and simply call it nonsense. "Gay people should have rights" and "gay people shouldn't have rights" are not opinions of equal value, and it's not "eating your cake and having it too" to say that the latter opinion is unwelcome.
Second, there's a very clear relationship between the "left" topics you mentioned and the workplace; not so for the "right" topics. You can make an argument that abortion/contraceptives are work-related because of employer-sponsored healthcare (I think it's clear that employees should keep their noses out of each other's private medical matters, but... okay), but the rest are totally ancillary. If you want a "no politics at work" rule, then you wouldn't have conversations about guns (unless it's to advocate that your employer allow guns at work), you wouldn't have conversations about immigration policy (unless it's to advocate that your employer stop sponsoring visas), and you wouldn't have conversations about furries (unless it's to advocate that your employer ban furry sex in the workplace... which I don't think is a problem that anyone has). You know what you would still have? Conversations about discrimination in the workplace. Conversations about whether your employer's political dollars are funding violence. Conversations about whether your recruitment policies are unconsciously biased.