I can't answer the question, but can offer a contemporary dissenting perspective.

Around 1600, George Silver wrote a couple of books - Paradoxes of Defence and Brief Instructions on my Paradoxes of Defence describing his take on the English tradition of martial arts.

Silver was of the opinion that the trend towards thrust-only weapons was a Bad Thing, and recommended what he called the short sword - what we would know as the basket-hilted broadsword.

He had a lot of disparaging things to say about rapiers, rapier masters, and rapier technique.

One of his comments stuck with me - roughly, that the rapier was useful only in duels, but that the broadsword was equally useful in war and duelling. He believed that it was pointless and irresponsible for someone to train in the art of arms, but be unable to use that training to defend his country.

So, that's the historical perspective of someone who thought the move towards rapier-family was a mistake. The rest of this post is from me.

Something to remember is that there is more to what a sword is good for than simply whether it's a cut, thrust, or cut and thrust design. A rapier, despite being a thrusting weapon, has a very flexible blade, and that makes it particularly ill-suited to dealing with armour. It very much is a sword designed to be used only against unarmoured targets.

Contrast with the estoc, which is a superficially similar weapon, but has an incredibly stiff blade, and is intended primarily as an anti-armour weapon.

Edit: I'm sad that this is the top comment! I was hoping for more good info about the evolution of the rapier...

John Clements of the Association for Renaissance Martial Arts writes the following:

"Rapiers developed from earlier forms of cut-and-thrust swords as a weapon for urban self-defense and private dueling. By the 1540s, swordmakers were responding to the self-defense needs of unarmored fighting men for a fast single-hand stabbing sword with a long reach that could be used in the street or back alley or an enclosed space. Following the ancient process of feedback between makers and users, their creations were tried out, and whatever worked best was then continually refined based on the advice and requests of swordsmen."

Your assumption that they evolved to counter the increased use of armour is incorrect, as /u/hydrogenjoule already pointed out. Indeed, the espada ropera that you mention more or less means "dress sword", i.e. a sword worn with civilian clothes as opposed to a heavier battlefield sword.

This article from Historynet.com appears to imply that a large part of the rapier as a thrusting weapon is the result of techniques developed by a number of Italian fencing masters.

I think you may be confused. Thrusting attacks were better against armor than slicing attacks, but that does not automatically mean the opposite is true against unarmored opponents.
Puncture or stab wounds are always a more effective than cutting/slicing hits because a puncture would is more immediately disruptive to the body than a cut. The difference is that an attacker will usually continue to fight, even though cut, until blood-loss finally has an effect. Edit - "in armored" = "unarmored"

To understand it you have to take a step back: what caused the evolution from the longsword to the sidesword?

The answer is your opponent's armour

Now, when you use a sword against an armoured opponent you need to pass his armour. You can try to hit the weak spots (which is dangerous, because you need to be slow and near him, and then he can do the same thing to you) or you can hit hard enough. Hitting requires force, which you know is defined as mass per accelleration. Now, a human body can develop a limited accelleration (expecially if you don't want to be able to swing your sword again, which is exactly what you hope to do) so the solution is to make heavy swords.

But heavy sword are uncomfortable and necessary slower than light ones. Also, the original longsword was without any finger protection and , I assure you, hit your opponent's fingers is easy (and it's painful even with blunted weapons, I don't want to imagine how painful it can be with a sharp blade).

So, you have something heavy and with little protection, what's the first thing you do? You add protection. You start with rings, and go on. While you are adding protection for your hands some chinese guy finds out a little powder which goes "bang!" and do awful things to people, even if they are in full armour.

Now people start wearing lighter armour, because the main danger simply doesn't care about how much steel you are wearing. Also, some clever smith finds a way to build thin blades which don't break. Now you don't need this heavy and unprotected sword: it's easy to pass your enemy's armour even with a light blade. Maybe you even fancy the idea of having a secondary weapon (a gun, a knife or even your fist if this is what suits you). If you remember the shape of a spada da lato this should look familiar to you.

Now, with a light blade cuts are less effective. You hit with less force, so you do less harm. But a thrust is a thrust, and is still effective, even with less mass on you sword (you can always put your own weight with it), so thrusts become more important then cuts.

To thrust you will need less mass and, by the way, this little powder is making more and more useless wearing an armour, so you will need even less mass! Your weapons becoming quicker, hitting the hands of your opponent become easier, so he put up more and more protection rings, until his smith tires and puts on half a sphere of full steel. Obviously you copy the idea and you find out you are holding a rapier.

Long story short: with lighter armour to hit you will need lighter sword, which are more effective in thrusts, which need lighter swords. In the end, this led to the smallsword.