These groups can lose all those rights, but it takes crossing a certain line.
Many years ago, a similar group operated here. They were loud and numerous, especially at certain events. One year, one of them slipped up and swung a sign at a peaceful citizen. That was the end of it. Restraining orders, loss of the right to protest, gather in public spaces, or give public speeches. They had to fight hard to continue operating as a "church" within city limits.
Without the hate agenda, the congregation dissolved in a fairly short time.