You have stated, "Modern translations omit John 5:4 so please avoid mentioning it". Don't you want to know why modern translation omit it? Has your mind been fully made up about the need to refuse this verse any consideration? Yet you ask this Q in a hermeneutics site, which requires examination of text verses to see if the text itself can provide any clues. Alas! The sometimes-missing verse 4 is the only one in the Bible that offers any explanation! And you have ruled it out of court at the outset! Some Greek texts, like Westcott & Hort's, and Nestle's, don't have the verse, but others do.
This only leaves answerers who agree with such a dismissal to speculate, to surmise, and to guess. That's hardly a sound hermeneutical approach.
Well, let me go along with your request, to demonstrate some ideas about why the waters could have been stirred up periodically, with no reference to angelic miracles. (You do realise, I hope, that that should logically include refusal to acknowledge Jesus as performing his miracle of healing at the pool-side? But those who agree that he did a miracle yet also think that Jesus was first created as the first angel, have some explaining to do.)
John's gospel (the last one to be written) is the only one to mention Jesus' healing at this pool. John deals with Jesus' miracles in Jerusalem, while the others dwell mainly on his miracles in Galilee. Jesus had gone up for the feast, and the city was very full. If verse 4 is ruled inadmissible, we only have verse 7 with the crippled man's own words to Jesus to go on: "...when the water is disturbed" (or, stirred, or troubled, as it is variously translated). There is no clue as to why this occasional movement of the water was observed. Jewish writers make no mention of any healing associated with the pool of Bethesda, though arguments from silence don't amount to any proof. Speculation, then:
"Some conjecture it began when Eliashib the high priest began the
building of the wall about Jerusalem, and sanctified it with prayer;
and that God testified his acceptance by putting this virtue [of
healing powers] into the adjoining pool. Some think it began now
lately at Christ's birth; nay, others at his baptism. Dr. Lightfoot,
finding in Josephus, Antiq. 15.121-122, mention of a great
earthquake in the seventh year of Herod, thirty years before Christ's
birth, supposed, since there used to be earthquakes at the descent of
angels, that then the angel first descended to stir this water. Some
think it ceased with this miracle, others at Christ's death..." [note
the earthquake when the angel descended to roll the stone away from
the tomb. Woops... I've mentioned angelic activity.] (Matthew Henry's Commentary, p1550)
Such ideas, if held by various Jewish people over the centuries, could account for them attributing miraculous healing events to the waters when they were stirred up. For those who dismiss the very idea of invisible angels occasionally causing the pool to be disturbed, then recourse must be made to supposing the pool nearby Bethesda may have had sluice gates opened now and then, causing a rush of water in, from underneath. It sounds similar to those who dismiss divine purpose in the parting of the Red Sea to let the Israelites through, explaining it away as a great east wind, or that the crossing point was very shallow etc.