Although Cicci was wise to pat Tess down as a precaution, Mafia caporegimes seldom carry because they're obvious targets for law enforcement, and if they're caught carrying, they go right to prison without The Law having to prove anything else. So, they usually don't carry--but travel with bodyguards who do.

I think Barzini also would have had Clemenza killed at the same time. He'd have kept Tess on for a while to organize the remnants of the Corleones. But once a traitor, always a candidate for being whacked.