I think you're overestimating Gladbach: the club has made it into the CL only three times, this was the first time they made it into the knockout rounds, they are working on a limited budget, even compared with German (top) clubs and before Covid they only averaged 50k in attendances. The club probably has some room left to grow, but it's not a sleeping giant in the sense that Dortmund were before Klopp or maybe Hamburg/Schalke are. They probably aim for top 6, especially given Leverkusen's level under Bosz. I mean sure, if he feels like he has to leave the Netherlands then a lack of better alternatives might make him consider Gladbach, but their squad isn't built for possession football: Embolo, Hofmann, Stindl, Thuram.. they all want to play direct football and a lot of teams in Bundesliga love nothing more than to play against a sloppy possession side. When the Rose rumours to Dortmund came up Kohfeldt was probably the most commonly named replacement candidate. Marsch might also be a logical choice.
Nagelsmann already stated, with regards to Dortmund rumours, that he's not leaving Leipzig next summer and anyway: "Rasenball's" philosophy is not possession football either.
I don't think that's a good idea either. Hasenhüttls football is a natural fit for an underdog side, but ten Hag made his name coaching a top club, tactically he'd have to make steps back and compromise even more than he would at Gladbach and we all know how eager the British press is to proclaim some foreign approach as unsuited to the PL.