Yes, if he really wanted to spend more time on this compared to other aspects then time would be found. But how about a theory that he's instead focused on things like defensive organization, attacking movement, pressing and integrating new players that has helped us to 12 points from 5 games despite not scoring from a single corner? My point is that corners are only one of many areas where improvement could be found. Training ground exercises aren't a zero sum game, but if he had spent an additional 2 hours per week on set pieces (picking a number out of my bum) then this would have been 2 hours less for other areas of focus.
I cannot show you any specific evidence for attacking corners at this point. Although I would like to point out at the same time that you have no specific evidence that we haven't worked on it.
Yes improvement is possible, but is an improvement from corners the most important at this time is a different (and probably unanswerable) question?
We haven't said all details, have we?
Honestly, we have people at the club whose job is to analyze games, results, performances, statistics and so on. You don't think they've been able to identify this corner thing? Really? I just don't see how that would be likely, if even possible.
You have given examples where you think what he's done hasn't given the (to you) desired result. That doesn't mean that the problem wasn't identified or that attempts to sort it out weren't made.
(As an aside I disagree completely about Townsend. Had he been kept around as a squad player I think he would have remained at the same level he was before going out on loan, the loan really improved him and we now have a very useful squad player on our hands at least)