How to politely tell a colleague they won't be an author of my article?
TLDR: Unless explicitly discussed otherwise, your friend does not expect to be a co-author. Just write a friendly email thanking them for the discussions and asking them if it is ok to include them in the acknowledgements.
As pointed out in the comments, this highly depends on which field we're talking about. In some fields, even pointing in a direction that actually turned out to be right might only be worth an acknowledgement (probably specific though, "we thank X for point to the results published in <obscure jounal 60 years ago>"). Since this is clear to most people, based on the information you provided, I would not think that your friend even expects to be co-author at all (unless they explicitly asked to be co-author, which would be border-line in terms of academic integrity but happens when the person urgently needs another co-authorship for formal reasons - in that case, however, one would expect that person to (a) communicate this clearly beforehand and (b) do much of the mindless busy-work, e.g. beautification of images).
That being said, I would assume that your friend knows they're not going to be co-author and thank them by email for the interesting, albeit ultimately fruitless discussions (but hey, maybe we can explore this in the future,...) and ask them if they agree to be mentioned in the acknowledgements. In other words, put a positive spin on it.
We made good progress pursuing direction X, but, it didn't work out due to Y, so we pursued Z, as described in the attached. We've included an acknowledgement of your input. Thanks again.
Under the circumstances, I doubt your friend expects co-authorship of the paper. Your idea of sending a courtesy email to let him know of your proposed decision (instead of having him read the paper and see he is not on it) sounds like a good idea to me. I recommend you write a courteous email thanking him for his help, letting him know that you and your colleagues discussed the issue, and letting him know your thinking on the matter. Do not frame this as a decision that has already been made --- instead, offer him an opportunity to respond before you and your colleagues make your final decision, in case he disgrees with your views and wishes to make an argument for his inclusion. Do not invite co-authorship unless you intend to follow through on that.
It is also worth noting that you could give your friend an acknowledgement in the paper rather than co-authorship. If you decide that this is appropriate then you could offer this as an alternative to co-authorship. If your description of events is accurate, the most likely response is that your friend will be happy to have had a chance to hear about the matter before seeing the published paper, and most likely he will agree that he has not done enough to warrant co-authorship. In the unlikely event that your friend insists that he deserves co-authorship, refer the matter for decision to your Head of Department or some other third-party for an independent decision.