It was long ago, but I recall that in my initial training, because of local conditions, we all 'happened' upon cross-wind landings. It wasn't planned that way, but when a x/w situation appeared, we had it pointed out to us what was happening to the a/c, its cause and the remedial action we should take. This was all within the pre-solo stage of our training.
So it came as little surprise to me when my instructor one day after a full stop landing said "OK, your turn to do it on your own", and vacated the cockpit. I do recall that the following take-off, circuit and landing seemed the longest in my career.
As far as "not getting it quite right", that is possibly the best form of instruction that there is. Once you get good at this flying stuff, you start to get a little cocky, until, one day, something bites you when you're least expecting it.
As far as teaching x/w before first solo, horses for courses. Some ab initios handle it well, while others don't. It's a good instructor who can read the signs in the student and instruct accordingly.
But, at least in principle, I agree that instructors should remove as many of the variables as possible leading up to that all-important first solo. Sometimes, though, x/w is a variable that cannot easily be removed.