Is it possible that the rapier started off as a cavalry sword? I am thinking about this for a few reasons:
1. Reitschwert, meaning riding sword is used synonymously with sidesword, which is a modern word for a rapier that is less rapier-like. Reitschwert can also describe longswords, and I don’t know what makes a long sword a Reitschwert or not.
2. Complex hilts allow better finger protection, allowing one to wrap a finger around the cross and give point.
3. The Polish Koncerz is similar to a rapier, especially a "true rapier" in that it is a long thrust dedicated weapon. In fact, it is longer and more thrust oriented.

I am not saying that rapiers weren’t duelling weapons, but rather wondering if the original idea was similar to the idea behind the koncerz.

Not terribly likely I think.

Terminology is mostly post-hoc, so I wouldn't trust the "reitschwert" thing too much.

Finger rings are actually first documented on cutting swords, before being adopted on other swords. More hand protection is always good, of course, but I don't think the fingering actually has so much to do with thrusting (mark that the koncerz do not have finger rings, either).

I'm not saying rapiers weren't used on horseback, of course. But they don't seem to have been designed primarily for horseback use. Perhaps the strongest piece of evidence for this is that in all lands were they were imported as foreign weapons (England, Germany essentially), they were described as dueling focused and on foot. I'm not aware of any source text alluding to a horseback development.

Regards,

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One problem is that complex-hilted military swords with four-sided blades often get described as 'rapiers' today. Seventeenth-century swords like the "Munich Town Guard" group or these in Schloss Ambras are not rapiers as described by George Silver or as depicted in the prints to Thibault and Capo Ferro. They are shorter with wider fortes.

The Met even calls this a rapier even though it has a blade that would look fine on an eighteenth-century English backsword! https://www.metmuseum.org/art/collection/search/22366

Reitschwert is more of a historical term whose meaning is unclear today, rather than a new creation that represents a clear typological distinction. It is possible that a few of the museums that use the term Reitschwert, have taken the term from their inventory. Wendelin Boeheim doesn’t mention the term, using the term Degen for rapier. He uses the term rappier to refer to a flexible practice weapon.

When checking Boeheim I noticed that he mentions that the Degen was used as a cavalry sword, and continued to be used as such in Southern Europe after it was adopted by foot soldiers in other parts of Europe. His definition of Degen includes Haudegens, which he defines as single-edged Degens.

The word Degen at some point became a synonym for sword, but I will have to check when. It is possible that in German, the word Rappier changed in meaning significantly by the time the Boeheim was writing.

If the rapier started as a cavalry weapon, you'd expect to see its length shorten when it gets adopted by infantry. We see the opposite occur; rapiers get longer over time before shortening again.

Many swords with a lot of rapier characteristics have blades that could easily be mounted on a late medieval longsword in the type XV, XVI, or XVII families except for the ricasso. And those longswords could easily be called tucks or estocs even if they are not the most extreme. So its not hard to see why in sixteenth-century England "tuck" and "rapier" can refer to the same object even though not all rapiers are suitable for pushing someone out of the saddle from a moving horse and not all tucks are pretty and easy to wear around town.

I don't know of a study of sword blades from sixteenth-century Europe because they are hard to date and hard to fit into a typology.

The Met has photos of two sixteenth-century estocs. You can see how someone looking at the Saxon estoc in its sheath might not know whether its more for poking at the gaps in plate armour, or impaling people wearing doublets and hose.

https://www.metmuseum.org/art/collection/search/25627
https://www.metmuseum.org/art/collection/search/27434

Edit: a longsword-hilted estoc with a blade section that reminds me of a smallsword https://philamuseum.org/collection/object/211938

Long narrow stiff blades first appear on swords for fighting armoured opponents, but a lot of early complex hilts are mounted on broad flat blades, and eventually some cutlers start to offer swords with both features.