Blundering into it doesn't make one automatically bad at whatever the job is, or a bad person. I do agree talking to the inmates is dangerous, also the temptation of trying to harness the power in that apparently very powerful ley line. However in the light of all of that, why didn't the Senior Council make sure there was a Warden in charge of the place they could trust? After all they have a designated assassin don't they? Someone authorized to use black magic for cripe sake!
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We don't really know, but we can speculate.
For one thing, the position of THE Warden is fraught. I suspect that it's kind of like appointing a new Blackstaff, only worse. The Senior Council would want to pick someone with serious chops, it's not the sort of job you'd want to give to a newbie (like Harry!). Also, you'd want someone stable, and trustworthy, and at the same time maybe not one of the oldest, most powerful Wizards either. You'd want to know that you could take him out if you need to.
And presumably whoever the SC picks has to be acceptable to Alfred, too.
So it's going to be a tricky pick. They'd probably prefer to leave the job empty rather than risk appointing someone they didn't trust...and we should remember that Wizards live for centuries and the SC are usually among the oldest of them. The Council, as an organization, is almost 2000 years old in its current form, and older yet in previous incarnations. They take the long view.
To us, the job of THE Warden (or the Blackstaff, or any position except maybe the Gatekeeper) being unfilled for 25 or 75 years might seem like a long time. But to the Senior Council it might seem like a modest interim while they pick a new Warden suitable for the job.
Probably most of the time, they'd be right. It's just that right now the world is entering one of the Wizard crises that come along every now and then.
If the island was Wardenless from 1961 to 2009 (assuming that was when Harry took over), that's a gap of less than half a century. The Council might very well have been in the process of vetting possible replacements. If Kemmler was the last one, that would make them doubly cautious about picking the right Wizard for the job, I'd tend to think.
Ebenezar's writings suggest that he thinks Harry might have been a good choice if he was older and more experienced, but he's not sure. He saw how Margaret started out promising and went off the rails, and there's a lot of his mother in Harry.
Langtry, OTOH, would probably have been passionately opposed to the idea of Harry as THE Warden. "One Kemmler is enough."
A lot of the stuff the White Council, and esp. the Senior Council, does makes sense if you remember the time perspectives they operate under.