Which sounds fine, but Woolworth's chose not to serve Black Americans. This was legal[0]. With the benefit of hindsight, we can claim that refusing to serve Black people was indeed "taking a side", but there were places where supporting the pro-integration people was breaking the law. And refusing to serve Black people, until they staged sit ins, didn't appear to be taking a side, because it wasn't uncommon. It was the way most places worked.
Even the mission to "serve everyone a sandwich" doesn't require addressing segregation, you can have separate lunch counters and serve everyone. Perhaps that's fine, people at the time certainly thought it was.
[0]: https://www.history.com/topics/black-history/the-greensboro-...