I could be wrong. But I'm not convinced with what you've offered as evidence.
You are technically correct, the President signs off on all appointments. But it doesn’t mean he chose him.
And frankly, even if he did, being chosen by Reagan to an AG job is not an endorsement by Reagan of anything other than your ability as a lawyer (or, as I believe, the opinion of the person who told Reagan to pick you about your ability).
We had this argument last year in Virginia with lefties who wanted to claim Ronald Reagan would endorse James Webb because Reagan said nice things about Webb when Reagan put him in as Secretary of the Navy. So we are very familiar with opponents misusing Reagan for their own nefarious purposes.
In that case, Nancy Reagan complained about Webb using Reagan in such a crass, political way. But Webb still ran a commercial with Reagan’s public speech about Webb back in the 80s.
I presume Rudy won’t be that opportunistic, and hope his supporters can stop misusing Reagan (who is dead and can’t defend himself) by pretending that Reagan would support Rudy in any manner.
That picture is more likely from his appointment as Associate Attorney General (#3 spot in DOJ) in 1981, not his appointment as U.S. Attorney for the SDNY in 1983. The Assoc Atty Gen position did require Senate confirmation (I found articles from March 1981 in the NYT referencing the confirmation hearing)