How can you tell that something "can exist without any ambiguity within it", if it "is undefined, or even cannot be defined"?

And what constitutes being "still a tenable concept"? What exactly are you trying to defend these undefined terms from?

Your examples seem less than convincing.

• The natural numbers: The existence of undecidable statements in formal arithmetic seems to me to indicate that there's more ambiguity about the natural numbers than we might like to think.
• Sports: OK, let's stipulate that we can nail down the definitions of (American) football, soccer, and basketball well enough to eliminate ambiguity. Haven't you then also established that "sport" in this sense is not undefined, still less indefinable?
• Free will: Different people mean very different things by "free will" and (ironically) what you've written here has not made it clear to me what you mean by the term -- but it seems clear that even you aren't convinced by your own example since you say no more than "I think it would deserve the name".

I have the impression -- perhaps quite wrongly -- that there's some particular example or class of examples that you have in mind but haven't listed here, which provides the actual motivation for this. Certainly the examples you've given don't seem to me like cases where (1) a concept is undefined or even indefinable, (2) it's clear that it "can exist without any ambiguity within it", and (3) there's any call to defend it from charges of untenability (whatever that actually means). Am I wrong?

(It occurs to me that what you've written would be more credible to me -- though I think I'd still disagree -- if by "ambiguity" you meant not "multiple or indefinite meanings" but something more like "internal contradictions". That seems like an odd misconception to have, but such things can happen to the best of people. The only purpose of this paragraph is to give you a chance to notice and clarify in the unlikely even that that's what's going on.)