This passage is a new detail to me, but Internet searches seem to indicate that although there was a pool there discovered in the 19th Century, that the passage you mention only appears in some manuscripts, and that the passage about the angel may be a later addition by someone.
“Verses 3b-4 are not found in the most reliable manuscripts of John,[5] although they appear in the King James Version of the Bible (which is based on the Textus Receptus). Most modern textual critics believe that John 5:3b-4 is an interpolation, and not an original part of the text of John.[6]
In these lay a great multitude of impotent folk, of blind, halt, withered, waiting for the moving of the water. For an angel went down at a certain season into the pool, and troubled the water: whosoever then first after the troubling of the water stepped in was made whole of whatsoever disease he had. And a certain man was there, which had an infirmity thirty and eight years (Interpolated text in bold).
The New English Translation and the English Revised Version omit this text completely, but others such as the New International Version refer to it in a note. )
And before a pool was actually found, the passage was taken to be entirely metaphorical:
“Until the 19th century, there was no evidence outside of John’s Gospel for the existence of this pool; therefore, scholars argued that the gospel was written later, probably by someone without first-hand knowledge of the city of Jerusalem, and that the pool had only a metaphorical, rather than historical, significance.[1] ” (https://en.wikipedia.org/wiki/Pool_of_Bethesda)
I have also found notions that the water contained therapeutic elements (magnesium), and that the pools there were used by the ancient Romans as a healing pool, ascribed to their gods Asclepius and Fortuna.
It’s striking to me that a Christian can not see how any or all healing and salvation in the Bible would not be primarily of metaphorical and spiritual significance rather than literal factual history for the sake of information.
I don’t think I’ll be able to learn the intentions of whoever added that passage, but I would think that the writer meant it as an actual angel… whatever that meant to them, but that probably the intention was more metaphorical and spiritual than literal.
But I would also say, both for the original writer’s intent, and for modern interpretation, that I’m not sure there’s much to add by trying to make the distinction, and to do so might seem to me to be aiming off the mark.
I don’t follow why you’d say an angel going down and stirring waters would ever be something other than doing God’s bidding. For isn’t God and his will supposed to be omnipresent in all things?
But if it helps, I would not think the intent is to say there’s a magic healing happening because an angel takes it upon himself with no influence from God to go stir water which happens to cause healing but has nothing to do with God. Again, I don’t know why you’d think that except it seems like modern thinking on cause & effect trying to interpret the Bible literally.