There's a difference between trying to attract players from Southampton and Newcastle to join your project than there is signing a top player from one of the few clubs in the world that are above United's level. Mane had nowhere near the reputation FdJ did at the time of his move either. Also, you have to look at trajectories. United were a top team that are in steep decline. Liverpool were a fallen star that hired a top coach and had a SERIOUS footballing department behind them that are cutting edge in almost every way.
This part is speaking in hindsight. Liverpool when Klopp joined was as much of a shamble as us. What was the proof of this "cutting-edge footballing department" back then?
They signed Firmino as a "blockbuster" to counter our Memphis deal, and did not even know how to use him properly, hence Firmino flopped in his 1st season. It was Klopp who resurrected Firmino's career at Liverpool. Don't get me started on their panic deals like Benteke who quickly showed he was not suitable for top-class clubs (As many people had already seen if they watched Benteke before). They were a one-man army in that almost-win-the-league season and all they did with the Suarez money after that was raiding Southampton for the likes of Lallana and Lambert. I see no sign of this supposedly cutting-edge footballing department you mentioned. It was Jurgen Klopp who completely elevated them. They were not like City who had a very good structure in place and only waited for the final figure (Pep Guardiola) to complete their organization.
So moving on from that, you mentioned we are in a different trajectory than Liverpool back then does not really have proof. Liverpool was a fallen star that won the lottery with Klopp. We are a fallen star (with more money than Liverpool) who got Erik ten Hag. Time will tell if he is a similar jackpot.
You had some points about Mane did not have FDJ's reputation. But we have the past working relationship between ETH and FDJ to take advantage of instead. The profiles of Southampton/Newcastle and Barcelona are really different. But it's a moot point because this is the case of Barcelona WANTING to sell. If Barca are determined to keep FDJ, then that's an entirely different story, though.