Gate staff should prevent drunk pax boarding in the first place
They do, certainly they do their best. Jet2's gate staff are empowered to make those decisions and we don't even see it on the aeroplane, rather than in years gone by where the gate staff would ask the crew, they don't any more, they just offload them.
they are also the carrier with a bigger problem than other budget airlines, not something to be shouting about.
I would disagree that Jet2 has more, just more publicity because Jet2 is making a stance and is fighting back against this small minority's. Perhaps if other airlines took the same firm no nonsense approach you'd see it is industry wide, irrespective of the carrier, be it low cost, tour operator or 'high end' national flag carriers.