Commands that don't need a target don't work in execute

Commands like:
/execute @p ~ ~ ~ tp ~ ~1 ~
don't work correctly (teleporting the nearest player one block up), but instead give out two error messages:
Player 'commands.generic.player.unspecified' cannot be found
and
`Failed to execute 'tp ~ ~1 ~' as <Player name>
where <Player name> is the @p player.
There are three bugs here, but if one gets fixed, the others become irrelevant:

• The command doesn't work like it would if it got executed from the player (which execute should simulate).

• The player name doesn't get put into the first error message. If the target selection for example meets an item, it gets displayed the same way as in the /kill feedback, but not for the player.

• There are two error messages for one error.

If the first one gets fixed, the other two get irrelevant.

Other examples are:

• /execute @p ~ ~ ~ scoreboard teams join test

• /execute @p ~ ~ ~ achievement give achievement.openInventory

• /execute @p ~ ~ ~ clear

• /execute @p ~ ~ ~ gamemode 1

• /execute @p ~ ~ ~ kill

• /execute @p ~ ~ ~ setworldspawn

• /execute @p ~ ~ ~ spawnpoint

• /execute @p ~ ~ ~ xp 1

Special cases:

• /execute @e ~ ~ ~ help shows the help to the player who executed the /execute command once per entity, not to the selected entity itself.

• /execute @e ~ ~ ~ seed shows the seed to the player who executed the /execute command once per entity, not the one selected entity itself.

• /execute @p ~ ~ ~ msg @p test shows _<Player name> whispers to you: test_ and _You whisper to <Player name>: test_.
  /msg @p test shows the error message Player 'commands.message.sameTarget' cannot be found. The same error message appears for every other entity with its UUID as player description.

• execute @p ~ ~ ~ publish works correctly.