Mowers will only mulch properly over a very finite height range and that range will depend upon the airflow characteristics of the deck & blade system.
In the USA this is made even harder as swing back blades on discs are considered too dangerious so you are relying on a 2" wide strip of steel to suck the grass up so it can be cut then twirl the clippings around, recutting them several times before depositing them back between the stems of the remaining grass, It is a real tall order. Take all the blades off your cars fan bar 2 and see how much air it shifts.
Despite the fact they put 6 to 10 height settings on mulching mowers, most will only mulch properly on the top few settings.
Then within the mulching height range of your mower, there will only be a couple of settings that will work with your particular grass type & density.
On top of that, the chopped up grass has to have somewhere to go thus the 1/3rd rule and in reality 1/4 would be closer to the mark if you want the clippings to vanish without trace.
Mulching requires a lot more energy than just cutting which is why the mulch only mowers usually have the largest engines.
It is a problem I get presented with almost daily but people just can not seem to understand mulched grass has to be cut no lower than 2.5".
If you want it lower than than then you need to get a reel mower and mow daily like they do on golf courses and bowling greens ( do you have lawn bowls in the USA ?)