Background Pony #6A14
This is a good question, lets weigh the strengths and weaknesses.
Alicorns are of course op compared to any other race but due to there only being 5 of them they are doomed to win battles but lose the war.
Earthponies are strong and resilient and good farmers which are all good perks but I don't think they are enough to make up for their lack of magical abilities.
Unicorns have their magic going for them and they would likely have a small number of unicorns strong enough to go toe to toe with an alicorn (Like Starlight Glimmer as seen in the show) but the average unicorn is of course much weaker, I would say they are the second most likely race to win (teleportation, cloud walking and turning your foes weapons in to teacups is very helpful) but I see the last remaining main pony race having even better chances.
Pegasi would be my pick for the likeliest winner, this comes down to two big advantages they have, flight and weather control, with these two combined they would be able to effectively choose where when and in what weather conditions to pick a fight and rain a mix of lightning arrows hail and spears on to their enemies stuck knee deep in mud due to heavy rains they created, heck they don't necessarily need to fight at all just send the worst weather towards your enemy and let disease and the elements cripple their armies. This would also deny earth ponies their farming advantage since the pegasi could kill their crops with either too much or little rain, too cold or hot weather or just a tornado trough their fields. They also seem to have a militaristic history before the unification of the 3 tribes. I think only unicorns, changelings and dragons would have a realistic chance against them but I'd say overall they would be the likeliest winners. the rest of the non pony races are too weak, too few in numbers or just cant counter the pegasi advantages.