I'm not sure if the problem is about the /tp or the /execute (but I think that it is about the /tp) but if you summon an armor_stand and then enter a command like this :
/execute at @p positioned as @e[type=armor_stand,distance=..50] as @p run tp @e[type=armor_stand,sort=nearest,limit=1] ~ ~ ~ ~ 0
The message tells that the armor_stand (which is supposed to take the orientation of the nearest player) is teleported but it doesn't change its orientation even if your orientation is different from his. No matter how many times you execute the command, it doesn't change its orientation but if you move your head and re-enter the command, it does and then, the problem disappears. I went in gamemode 3 and the armor_stand does take the good orientation after the first time the command is executed but it doesn't appear visually.

It's only a visual bug but which stays until you move your head left or right (moving it only up or down doesn't change anything)

Note : I tried this with a chicken and a husk, it worked perfectly, visually also. It seems not to work only with armor stands.