While launching versions between a1.2.0 and a1.2.6, the player name is always set to:
"Player" + System.currentTimeMillis() % 1000L
instead of the actual player's name.

This can be simply solved by adding a tweakClass minecraft argument from other Alpha jsons to these Alpha 1.2.x json files:
"minecraftArguments": "${auth_player_name} ${auth_session} --gameDir ${game_directory} --assetsDir ${game_assets} --tweakClass net.minecraft.launchwrapper.AlphaVanillaTweaker"

It's simple to fix, and if fixed, it will stop making people search for launcher alternatives just to join Alpha 1.2.x servers.