That is certainly an advantage, but you only have to look at old English warbow archers to see that a bow can be drawn a long way back on the usual side of the bow. The Mediterranean release, by the way, is the now-standard Western method of drawing the bow with three fingers, one on top of the arrow and two beneath.
For my own part, coming at thumb shooting from a horseback perspective, there are two advantages to shooting off your thumb side of the bow. First, it means that the wind of your passage is pushing the arrow against the bow rather than off it. If you put the arrow on your knuckle as if for a Mediterranean draw and then go to a gallop, you have to be careful that the arrow doesn't blow off your knuckle.
Second, it's quicker. Drawing from a hip-worn quiver (which almost all were), you can just pull the arrow out and put it straight onto your hand, nock it, hook your thumb onto the string and shoot. There is no need to start trying to thread it between the bow and the string as you have to for a Mediterranean side shot. The European 'Kassai' school of horseback archery use a Mediterranean release but it is noteworthy that they hold their arrows in the bow hand. This makes for fast shooting but is without historical basis and is not permitted under World Horseback Archery Federation rules. It is allowed only in Hungarian style events, which are also a modern invention not recognised by the WHAF.
All of which takes me neatly to the British Championships, which will be held in Vines Cross, East Sussex (just North of Eastbourne) on the weekend of 22nd October. There will be a Korean competition (the main event of the World Horseback Archery Federation, involving single shot, double shot (forwards and backwards in the same run) and a multiple shot (3 targets in 90m, requiring an arrow about every 2-3s)). There will also be a Hungarian competition (90m track, three targets together in the middle, so you shoot forwards for the first 30m, sideways for the next 30m and backwards for the last 30m, as many arrows as you can get off). There will also be a Hunt the Hare event (Britain's own specialist event, not WHAF recognised, in which a 3D hare target is placed on the ground near the track and the winner is the one who can hit closest to the centre of the target). There may or may not be a qabaq event (a circular target at the top of an 8m high pole, at which you shoot with blunts, twisting around and leaning out of the saddle to shoot vertically upwards, as shown in my avatar pic, although not very well because on that run a dog started to bark as I approached the pole and my horse got all skittish, so I chickened out of the the whole 'hanging off to one side of the charging horse' thing).
Should be a good event, all welcome. More details will follow as they are finalised.
Cheers
Dan