Just curious ... how do well-known agencies keep clear of LE, who *must* know about those businesses, and how do they protect clients' identities when something does go wrong?

Well known agencies get busted all the time. They survive because the LE doesn't bother to put them out of business. LE lets them be as it requires real effort to put together a case and win it.  The agencies have enough cash to hire lawyers and fight.  The LE can get the same publicity by busting streetwalkers who just plead out.

Agencies do not protect clients identities when their records are siezed.

Sometimes they can be a great source of Intel, so some police use them

Also you have left off GRAFT- many are actually owned by cops or ex- cops or folks who PAY OFF the cops in order to keep in business