The following is not more than a contribution to an answer because it could be that the garment referred to in the gemoro quoted is a talis godol and not a tallis koton.
The Gemoro in Menachos 41a brings a story of R. Kattina. This indicates that although the obligation of tzitzis is on the garment and not on the wearer, one is liable for punishment in a time of Divine anger for not fulfilling a positive commandment (wearing a garment that needs tzitzis) when it could be fulfilled.
For an angel once found R. Kattina wearing a linen wrap,(14) and he
exclaimed, ‘Kattina, Kattina, a wrap in summer and a cloak (15) in
winter, and what is to happen to the law of zizith?’ ‘And do you
punish’, asked R. Kattina, ‘a person [who omits to perform] a positive
‘In a time of wrath’, replied the angel, ‘we do’.
Now if you hold that the law of zizith is an obligation incumbent upon
the person then that is why one would incur guilt for not wearing a
garment with fringes; but if you hold that it is an obligation
attaching to the garment, then why [is any guilt incurred] seeing that
these garments are exempt? What then do you hold? That it is an
obligation incumbent upon the person? I grant you that the All —
Merciful would punish one who wears [without fringes] a garment that
is subject to fringes, but would the All-Merciful punish one who wears
[without fringes] a garment that is not subject to it?
— This is what [the angel] implied, ‘You find every excuse to free
yourself from the law of zizith’.
(14) Which was without fringes, since it was mainly used as a night wrap.
(15) A garment with rounded corners and so not subject to the law of zizith.
Shulchan Oruch 24a says that even though one does not have to purchase a garment to put tzitzis on it,
…..מ״מ טוב ונכון להיות כל אדם זהיר וזריז ללבוש טלית מצוייצת כל היום כדי שיזכור המצות בבל רגע …..
nevertheless it is right for everyone to be careful to wear a tallis with tzitzis to remember the mitzvos.
Igros Moshe O Ch chelek 4, simon 4 says that this is a Minhag Yisroel.
Rav Roei Siton points out that in Morocco the minhag was only for talmidei chachomim to wear a talis koton. The Minhag Yisroel is therefore not without exceptions.