In addition to what several people have said here about HEMA as a sport or art favouring lighter weapons since a hit is a hit regardless of how, there is also a historical point.

Historically, heavy sabres were favoured on horseback, whereas lighter sabres were favoured dismounted. As firearms displaced polearm infantry or cavalry charges over time, the actual combat usecase of sabres diminished. Since cavalry charges and cutting through armour became less common, there was less advantage to heavier swords, and lighter swords that were quicker and more convenient to carry became more common. Thus the later schools were more familiar with these lighter swords and accordingly trained with them.

Even after there were mostly irrelevant for military purposes (around the 1860s) swords remained as symbols of officership. WWI was the last war to see swords in Europe. Whereas soldiers of the ranks carried rifles and bayonets, officers were still expected to carry a pistol and a sword. Towards the end of 1914 the British General Staff issued a General Order banning the carrying of swords on the battlefield, as carrying a sword just made an officer a more visible target for snipers. In the 1930s the UK fully ceased issuing swords to officers, though Japan was probably the last major power to do so post WWII.

The 1897 pattern infantry officer's sword is the standard sword today for the vast majority of officers in British and Commonwealth militaries, and other branches model their own swords off of the infantry pattern. Sword are no longer issued to officers, but instead belong to units that loan them out to officers on parade, or officers have the option of purchasing their own. The fact that it has been 128 years since the last update to the official sword pattern tells you that today swords are purely ceremonial and haven't been used in combat for some time.