Butted mail was used. Some examples include:

• Parade armour. Reduced protection doesn't matter if the armour won't be fought in. Butted parade mail was made in India in the 19th century, often with patterns produced using a mix of iron and brass rings.

• Armour for the gun-dominated battlefield. When the main ranged threat is the gun rather than the bow and arrow, arrow resistance doesn't matter as much. Armour isn't worn to stop bullets, since it won't. If a common threat in hand-to-hand combat is cutting swords, armour can still be useful. Butted mail was used for cavalry armours in India in the 19th century. This appears to be an example of such armour: https://www.britishmuseum.org/research/collection_online/collection_object_details.aspx?objectId=545293&partId=1 Moro mail-and-plate armour is usually made with butted brass rings: (example 1) (example 2)

• Armour for other than the battlefield. If the main threat is cutting swords, and bows and spears are unlikely to appear, butted mail can work well. Potential applications include duels (if thrusting weapons weren't used), police armour, and anti-assassination armour. Such armour was used in many places. Some examples include Edo Period Japanese armour (example of gauntlets) and 19th century Caucasian (Khevsur?) armour.

• Japanese mail for arms and legs on battle armours often uses non-riveted links. These come in two kinds. First, some rings are a double loop, like a key ring. Second, some rings are thick butted rings. It's common for both types to be used on the same piece of armour, with the butted rings lying flat, connected by key-ring rings: (example 1) (example 2). Here is an example of Japanese armour with twin butted rings imitating key-ring rings: https://commons.wikimedia.org/wiki/File:Kusazuri_with_double_butted_6_in_1_kusari.jpg (it isn't obvious whether this is Edo Period parade armour the wearer never expected to fight in, or battle armour). While surviving examples tend to be from the gun-dominated battlefield, or the peace-dominated Edo Period (and thus candidates for the first two categories above), this style of armour pre-dates the gun's dominance in Japan.

It seems almost as good as riveted mail for most purposes

It is much less effective against arrows, and also less effective against other thrusting weapons. Stopping arrows is often a very important function of armour, and "most purposes" might not be enough if it doesn't include this particular purpose. This suggests that if archery is a significant threat, then you use riveted mail instead of butted.

It is effective against cuts, so it isn't useless.

Butted mail is often made with much thicker rings than riveted mail. If it's made with thin rings, it can fall apart under its own weight during movement. To reduce the chance of this, butted mail can be sewn to a backing (as seen in some of the examples above).