The verse is extremely unlikely to be original, and evidently a latter addition, as the NET Bible notes on John 5:3:
9 tc The majority of later mss (C Θ Ψ 078 ƒ M) add the following to 5:3: “waiting for the moving of the water. 5:4 For an angel of the Lord went down and stirred up the water at certain times. Whoever first stepped in after the stirring of the water was healed from whatever disease which he suffered.” Other mss include only v. 3b (A D 33 lat) or v. 4 (A L it). Few textual scholars today would accept the authenticity of any portion of vv. 3b-4, for they are not found in the earliest and best witnesses (P א B C* T co), they include un-Johannine vocabulary and syntax, several of the mss that include the verses mark them as spurious (with an asterisk or obelisk), and because there is a great amount of textual diversity among the witnesses that do include the verses. The present translation follows NA in omitting the verse number, a procedure also followed by a number of other modern translations.
The verse is clearly a marginal gloss found its way into the text, by the Byzantine scribes. It seems it must have been a tradition of the superstition known to those scribes even in the third and fourth centuries. It is apparently inserted to explain the verse John 5:7
[NET] The sick man answered him, “Sir, I have no one to put me into the pool when the water is stirred up. While I am trying to get into the water, someone else goes down there before me.”
There's no reason for the oldest manuscript families to have removed the verse as there is nothing embarrassing and problematic in it, to state the superstition or tradition. The only fact we learn from such later additions of interpolation is the habit of some careless scribes (particularly the Byzantine) to harmonize, trying to justify and edit the text according to their own minds and prejudices, and that it serves as a good example to figure out their habit of interpolation or corruption. While the information about the tradition of angel stirring the water might be true or a pure conjecture of the mind, the text deserve to be in the margins and commentaries.
To quote from Metzger and Ehrman's The Text of the New Testament, It's Transmission Corruption and Restoration, 2005, chapter 7: Causes of error in Transmission of the Text, p 258:
Errors of Judgment : Though perhaps several of the following examples might be
classified under the category of deliberate changes introduced for
doctrinal reasons, it is possible to regard them as unintentional errors
committed by well-meaning but sometimes stupid or sleepy scribes.
Words and notes standing in the margin of the older copy were
occasionally incorporated into the text of the new manuscript. Since
the margin was used for glosses (i.e., synonyms of hard words in the
text) as well as corrections, it must have often been most perplexing
to a scribe to decide what to do with a marginal note. It was easiest
to solve any doubt by putting the note into the text being copied.
Thus, it is probable that what was originally a marginal comment explaining the moving of the water in the pool at Bethesda (John 5.7) was incorporated into the text of John 5.3b-4 (see the King James Version for the addition). Again, it is altogether likely that the clause in later manuscripts at Rom. 8.1 "who walk not according to the flesh but according to the spirit" was originally an explanatory note (perhaps derived from verse 4) defining "those who are in Christ Jesus."
As was mentioned in Chapter 1, some manuscripts are provided with
marginal helps, designed to assist the reader of the fixed Scripture
lessons appointed by the ecclesiastical calendar (the Lectionary). As
a result, lectionary formulas, such as εῖπεν ό κύριος, occasionally
crept into the text of nonlectionary manuscripts (e.g., at Matt. 25.31
and Luke 7.31).