There are numerous reasons why light cavalry tended to go for curved blades more often, with traditionalism being a large part of it, but with draw cutting and recovery being an even larger part.

When you're on horseback, a thrust into an opponent can lead you to lose your sword = or at least be temporarily disarmed, as you'd normally use a sword knot to prevent the loss of your sword - whereas performing a cut is less likely to cause such a problem.

The "draw cut" refer to a slicing cut where the blade imacts the target, then moves across the target, "drawing" the edge across them. This occurs naturally when on horseback as the horse is already providing lateral motion. A curved blade makes this even easier, as they are easier to perform draw cuts with. A curved blade naturally passed over a target. instead of getting stuck in the target.

A draw cut has a lot of benefits in the early modern context you mentioned, as they're particularly good for getting through cloth and textiles. In the Crimean war, for example, many people using straight or only moderately curved swords noted difficulty getting through the heavy overcoats used - and some people using pipe-back swords were unable to thrust into their opponents due to excess flexibility - while the more heavily curved cavalry sabres employing a draw cutting doctrine experienced no such issue.

Now, that explains why a curved sabre was preferred in some cavalry context - now what about straight blades?

For the most part, the straight swords used around the 19th century come from the training systems used by various militaries being centered around the foil and the gymnasium sabre - one of which is straight, and the other curved.

In countries with lots of foil - based training systems, swords like the British 1796 Spadroon were adopted - a straight cut and thrust sword with a light blade, and smallsword-like hilt. Likewise, the french infantry officers of the same time period used either a courtsword - which is essentially a smallsword, but with an asymmetric guard - or an epee du soldat, which is a smallsword with a double edged blade, instead of a triangular blade.

In branches of the military using gymnasium sabres more frequently, we tend to see light, curved sabres being used around the same time - for example, flank infantry and light infantry used a heavily curved, broad, lightweight sabre (the 1803) in Britain. This was also arguably a more combat effective sword than the 1796 british spadroon - which was often disliked due to many of them being overly flexible, though that's not true of all examples. The 1803 is an excellent draw cutting sword, but required very different techniques to the foil fencing a regular infantry officer received.

Because the regular infantry is such a massive component of the military in terms of numbers, and officers required, it made sense from a logistical standpoint to train line infantry officers on the foil, which was a commonly taught system to the upper class from which officers usually came.

Now, there are straight bladed cavalry swords which must be adressed. In the 19th century context, these were used by heavy cavalry regiments on both sides of the napoleonic wars - in the form of the 1796 Heavy Cavalry sabre in british service, and a myriad of different thrusting swords in french service, who were less standardised. Ther British weapon was a heavy, chopping weapon, not well suited to thrusting without regrinding the tip (which field orders issued on the night before waterloo ordered officers and troops to have done, in order to convert the hatchet point into a spear point). The french weapon, on the other hand, was a thrusting oriented sword.

These straight bladed cavalry swords go against all I've said about the importance of the draw cut, and the problems of thrusting in a cavalry context, and as such necessitated vastly different techniques. This was considered worthwhile to the french, who praised a thrust-centric doctrine, which was more likely to kill an opponent than merely injure him. It does make it necessary to either release your sword and reply on the sword knot to retain it, when passing a struck opponent, or to swing your arm behind you as you pass, which is preferred. Some heavy cavalry units at the time were still armored, wearing a breastplate and helmet, which may explain some of the preference for a straight thrusting sword if used in response to other heavy cavalry units - but that's a rare occurrence.

All in all, there are many reasons for sword designs, often being influenced by traditionalism and training doctrines. Straight blades and curved blades were used by both infantry and cavalry, and the nuances of which was to be used by which regiment require a much longer answer than this, especially on the french side where numerous patterns of sword were used by each branch.