The crux of the issue is many (including myself) don't believe Real Madrid wouldn't consider signing one of the best players in the world due to some misplaced pride, three whole years after this saga.
This is not just an issue of misplaced pride, though. If the story that the Spanish press are telling is true then Mbappe has shown to the club that he cannot really be trusted in a negotiation.
A 'best player in the world' with huge wages conditions all the planning, so if you can't trust they'll sign it's a problem. By all accounts RM are scrambling a bit now. Even if they wanted to sign him in three years it would be irresponsible to do anything other than send a "sign by 30th January or leave it" offer.
But then I would add that pride shouldn't be discounted. Perez likes putting on a show, Real Madrid is a circus. These big signings aren't just about the player quality, they're statements. The 'statement' of signing Mbappe is dead, he didn't move on a big money transfer, he didn't move on a scandalous free transfer, show's a dud. "The player is great" isn't enough.
Time will tell, I guess, but I can't understand where this confidence is coming from.
Confidence mostly from following the club's transfer dealings since I've been a fan. In my experience the transfers either happen in their time, or they never do.
For example Patrick Vieira tried to sign for RM in 2004, the deal was made between the two clubs, he agreed to join, but then backed away at the last minute. He left Arsenal the next season, but not for RM, he joined Juventus. Years later he said "If I knew I was leaving the year after, I was going to Madrid."
David De Gea was also an almost-signing, RM and United agreed on a deal, the fax machine thing happened, and then a month or so later De Gea signed a new contract with United and has stayed there the rest of his career.
Lewandowski, supposedly RM attempted to sign him before he moved to Bayern. After that it never really made much sense as far as the squad went, plus Bayern weren't interested in selling so it never happened.
On the opposite end you have Cristiano Ronaldo, where the main transfer season drama was in 2008. That ended with an agreement to sell him to Real Madrid next season for 90m. So in 2009 they announced his signing quickly, it was within a day of rumors I think. Had this agreement not been reached, he would not have played for Real Madrid, he would have (responsibly, intelligently) renewed his contract.
Like I said in another post, what Mbappe did with running down the contract was risky, "playing with fire". People make fun of Kane for signing 6-year-contract with Spurs and then moaning about leaving, but it's irresponsible for top level players to be dicking around with short-term contracts. I don't actually think that in three years, Mbappe will be a free agent. I think PSG gave him a short-term renewal so that they can convince him to do a long-term renewal, which he will sign because it's going to be the best option he'll have.