Re: "I am far from convinced that people in general wish to be seen as caring more about morality than they actually do."

Not everyone - just most people. Signalling you are moral and trustworthy helps win friends, partners and mates.

Re: "If this was the case, why would the persistent claim that people are -- and, logically, must be -- egoists have so long survived strong counter-arguments?"

Surely egoists want to signal how trustworthy and moral they are too - just like most other people.