My father fears and thinks that if he moves to another EU country, then he'll have to basically bargain his Italian PR for a temporary one in the other country
Your father is basically right, at least for the purpose of naturalisation. If he leaves the country and settles in another country, he would quite plainly not fit the residence requirement anymore and could be deemed to have abandoned his application.
EU law facilitates moving between country for long-term residents from third countries but member states are still completely free to define the requirements for citizenship. So he would not automatically be deemed to have resided long enough for that purpose anywhere. But as @phoog explained, he would still be a long-term resident under EU law, which should at least make it possible to come back to Italy.
Authorities of a country can ask in return papers issued by an other country
Do not conflate the document with the status. Another country is not going to ask him to surrender his Italian PR card and he might even be able to hide the fact that he moved from the Italian state for long enough and get away with it. There could also be a grace period during which his absence would not reset the clock for the residence requirement. But if he would effectively not be a resident of Italy anymore and it could backfire badly. That's a simple fact and in principle unrelated to having surrendered the plastic card itself.
He may not get Italian citizenship if he gives up is PR card (if that's the case), since his dossier is on process and he's satisfied all conditions pertaining to work and residence (there have been some controversies lastly with his case, because Italian administration oververified his case multiple times and they came out with the excuse that he's been unemployed for several years and then his average income went down, but after that I declared my income in order to stabilize things).
So it seems he has a weak application and is already under scrutiny. I think your father is right to be careful here. Especially considering the fact that once he moved and stayed away for 1-2 years, there is no coming back, he would have to stay the whole 5 years (or whatever count as permanent residence/satisfies the requirement to apply for citizenship in Italy) and letting this application fail would mean dealing with all these difficulties again.
Note that in some countries, even leaving after obtaining citizenship can be the basis for a procedure to remove it. The point being that the core requirement is not merely to hold this or that card, it's to live in the country and intend to continue to make it your home. I don't know enough about Italian law and practices to know if it's an issue there but going through the formalities to settle elsewhere would constitute prima facie evidence that your father does not intend to live in Italy.