If you ever find where Young worked out the equivalency to P.E. then you'll find he had m v² = 2 m g h. But I suspect he did not.
The underlying topic is vis viva. As https://en.wikipedia.org/wiki/Vis_viva points out, Leibniz was the first to point out that m v² was conserved, Bernoulli used 1/2 m v² in 1741, and at about the same time du Châtelet derived the notion of conservation of energy using Newtonian mechanics, Young called the concept "energy" in 1807 (though without the 1/2), and the 1/2 'recalibration' was due to Gaspard-Gustave Coriolis and Jean-Victor Poncelet during 1819–1839.
Quoting https://en.wikipedia.org/wiki/Gaspard-Gustave_de_Coriolis :
> In 1829, Coriolis published a textbook, Calcul de l'Effet des Machines ("Calculation of the Effect of Machines"), which presented mechanics in a way that could readily be applied by industry. In this period, the correct expression for kinetic energy, ½ mv2, and its relation to mechanical work, became established.
Coriolis gives the specific reason at https://archive.org/details/ducalculdeleffe00corigoog/page/n... :
> .. nous appliquerons cette dénomination à la moitié de ce produit, en sorte que la force vive sera le produit de la masse par la moitié du carré de la vitesse. Cette légère modification à l'usage ancien introduira plus de simplicité dans les énoncés des principes que nous avons à donner.
From Google Translate:
> we will apply this denomination to half of this product, so that the living force will be the product of the mass by half of the square of the speed. This slight modification to the old usage will introduce more simplicity in the statements of the principles that we have to give.
In other words, it's not "a useless convention" as you write, but a convention which simplifies the resulting mathematics.