I gather you are talking about Europeans and suchlike Western men.

The classical Greek and Roman tunics were way baggy. Belting them kept the fullness under control when you bent over or even bent forward. They didn't even wear trousers.

Trousers were for and from barbarians. Your Sarmatian horse nomads wore trousers, but also front-closing tunics. But they had no buttons. They belted their tunics closed.

Romans got to dressing their slaves like barbarians in trousers, but closed tunics. Also the whole tunics and trews look was Gaulish. They often used a drawstring on their trews but the belt served as a place to carry things because no one had invented pockets. Have a sheath knife? Hang it on your belt. Have a purse with coins? On the belt.

Fast forward to barbarian invasions, fall of the Western Empire, and everyone is wearing tunics and trews and hanging stuff on that outside belt like always.

This goes on to the Early Modern Period. One of the ways we know when we are is that men stop wearing doublets with belts with purses on them and start wearing coats and vests with pockets. The outside belt disappears, covered by the vest if it is worn at all. In most cases, breeches are held in place by their waistband that laces snugly in the back.

Belts don't disappear though. They are worn by farmers and other workingmen who wear socks to work. Just like the baggy tunic, the smock's fullness needs control and a belt is a handy place to stick a tool or whip.

By 1800, gentlemen did not wear belts, and hadn't for ages. They used suspenders when fashion decreed that breeches' waists should be above the pelvis.

The wearing of belts by men of some station seems to be an American habit first, possibly from the working wealthy, whether steamboat captains or miners who got lucky. In "Viva Mexico!" (1904) Flandrau, an Eastern gentleman, notes his cabinmate on the trip south has the annoying habit of wearing belt with suspenders.

Notice: belt is still over shirt, which is firmly tucked into trouser waistband. This continued thru the 20th century for the most part. What's required now is that men, bit by bit, stop tucking in their shirts. Fifties: wearing t-shirts and jeans, teenagers still tuck in. Sixties: dashiki shirts are never tucked in, but standard shirts are. We don't count shirts worn as jackets, like the surfer's Pendelton over tucked tee. Seventies: disco's French-cut shirts are too tight to wear outside.

So I would place the ordinariness of not tucking shirts to fashion of the 80s and 90s. It's not really about the belts.