I think it depends on who and where. I spent a while in Mumbai this year, and ate many, many batatavadas and vada pavs, and not a single one came with either of those. Not the ones I bought on the street, not the ones from old stalls (that one can order delivery from these days), not even the ones from our club. Just red (garlic) chutney.
Outside Mumbai and across the diaspora, the other chutneys may be offered because people who aren’t familiar with the original dish want “all” the chutneys, or restaurants just serve them that way. I have a feeling that the (Mumbai) chains that “blew up” the vada pav on a broader stage (like Jumbo King) as the Mumbai Burger might have had something to do with the extra sauces being offered as an option locally, and others may have matched to “compete”. No one had really felt the need to advertise or press themselves as “oldest” or “best” until then, at least in my recollection.
When we make batatavadas at home, we sometimes serve green (cilantro, not mint) chutney, but home vadas are different (thinner coating, less spicy).
(To your radish cake comparison, purist dim sum eaters in their origin place probably don’t douse their dumplings with all the sauces I do 
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