You're looking for a tool that's gonna do it all in one shot. That's not gonna happen. Cutting tall grass, especially with a rotary blade, takes far more power than you can generate via wheels/transmission on a push mower.
Really, that rotary push mower is a "finish mower" - it's designed to groom a lawn that is already in control.
You need to use other tools to get the lawn in control, first.
What I've always used is - gosh, it's hard to name it, but a "weed cutter", "swing grass cutter", "grass shear" or "swing cutter" or "swing blade".
It's vaguely like a scythe but double-faced, so you can be productive on both swings. They're crazy productive - once you use one, you won't be able to believe anyone uses string trimmers in open areas. Obviously it does take ...skill...
and this person sure doesn't have it... but once you get dialed in, it makes short work of the high stuff.
Then you bring in your finish mower.
If this doesn't appeal because you don't want to have to use it every 2 weeks, then the root problem is you don't mow often enough. Power mowers will let you get away with murder: you can cut stuff 2 feet high with power mowers. Not with a push mower. You have to be "on it", and pay attention to things that affect grass growth, like rain.