Well, I think you need a sense of proportion. The Carroll transfer is obviously a statistical outlier: it was clear to sensible people at the time that it was ludicrously overpriced, and that he was a flavour of the month player, and has proven to be one of the worst deals of the Premier League era. It's not that useful as an example of a particular approach.
I agree that the CL issue is undoubtedly a complicating factor, because the benefits of finishing 4th as opposed to 5th are so incredibly large. In theory it'd be worth paying out £20m for one year if the result was CL football. I don't think there are any easy answers to that particular equation, because as you say any money spent will always be a gamble.
And you can't persuade me that this approach is sensible just by stating it, as if it were a fact. Why is it sensible? And compared to what? My main point is that, if the owners are willing to pay £5m for one season's service, why are they not able/willing to pay £20m or so to buy a similar player outright? A 23 year old will be 28 in 5 years' time - therefore I think it's reasonable to estimate he could be sold for £10m at that point. The total cost in terms of transfer fees is £10m over 5 years, or £2m a year, less than half what we're paying for one year of Sahin. Would that not be better, if we were able to find such a player? Surely they exist?