Jardim isn't necessarily outdated, but I don't think you're going to see tangible improvement from him vis-à-vis Mourinho when you consider some of the things the latter is criticized for. For instance, the rudimentary nature of his attacking schemes where the team is just asked to counter and the success is predicated on individual quality and athleticism on the break (worked really well in Madrid for a bit — less so at Chelsea II and United, particularly when the defensive balance was disrupted), but Jardim is the same in a lot of ways — and even with peak Monaco, his inability to formulate diverse/intelligent attacking patterns and install a more exhaustive attacking scheme was exploited by Allegri in the Champions League, for reference. He's rightly lauded for Monaco's sizzling attack with Mbappé and Falcao and Silva and Lemar, but that was an anomaly for his career — a perfect storm where he gave in and the forwards ran riot with Mendy/Sidibé on the overlap. Apart from that high, his teams are generally quite cagey, and don't score a lot of goals for, either...if you consider some of his teams over the last decade:
1.8 goals per game at Braga
2.1 goals per game at Olympiakos
1.3 goals per game at Beira Mar
2.2 goals per game at SCP
1.3 league goals per game in his first season at Monaco, 1.5 league goals per game in the second
Even though José isn't in the best moment of his career right now, he's still a better manager than Jardim — United must avoid the honey-trap of employing a non-elite coach just to get rid of Mourinho...the grass isn't always greener, etc.