that = that the fountain of youth exists?

No, here "that" (the example of it which you have enlarged and bold-faced) is used as a conjunction to link "We accept" to and to introduce "as we age, our health and mental acuity will decline."

I think it's a relative clause used as a noun, the direct object of the verb.

"we / accept / that = health and mental acuity/ will decline

Try this to see if it helps make it clearer, Ori.

Professor Amdam says, "We accept that [because we know that the fountain of youth doesn't exist] as we age, our health and mental acuity will decline.

Got it.

Thank you both.