I have been reading recently that strength that is built with Kettlebells is highly specific to the Kettlebell lifts and is useless when the strength has to be demonstrated with a barbell or in a fight.
Not really. Kettlebells, because of their shape, require kettlebell specific skill to use them, and the shape does limit the weight that can be carried.
But, people who train with kettlebells are as strong as they can lift. If a person works up to 10 presses with a 24 kg kettlebell, then they can surely do the same with a 25 kg dumbbell.
The weight is weight, and the body does not know the difference. That is why kettlebells are not mythical tools for strength (so yes, they are overhyped a lot).
The consensus seems to be that Kettlebells are lighter than dumbbells so they cant build as much strength, and the position of the kettlebell on the forearm makes pressing “too easy”. I think that these opinions MUST be wrong; but I have almost no experience of pressing with a barbell. Who is right?
It is difficult to compare unilateral lifting with bilateral lifting. But, I found kettlebell pressing to be not easier than pressing a dumbbell of the same weight. It is a little different, but the greatest weight I pressed with one hand was a barbell (in a side press). I would imagine that the normal split is that a person can press with one hand 40% of what they can press with two hands (I am just guessing there), but it is also possible to be significantly stronger in pressing with one hand if one trains that way, and likewise, a person can be significantly stronger with two hands if they train for it.
In "real life", strength is less specific. I was
reading of a very good powerlifter who years later stated he was really only strong in the powerlifts. He couldn't do much else. So, judging strength depends on the context. Is a person who can squat a 1000 lb barbell stronger who cannot do a pullup stronger than a person who can squat 200 lbs, but can do a one armed pullup?
My warning about kettlebells is do them if you enjoy them, otherwise, it is a very specific way to train and requires instruction on technique just to handle the shape of the weight properly.
I would suggest, in addition to your bodyweight training (calisthenics is the first and only essential way to train, if a person cannot move their own body well, they are weak) that you try heavy barbell training, maybe StrongLifts or 5/3/1 or something like it.