Why isn't the Mythos phase the fourth phase in the turn sequence rather than the first? That way you wouldn't have to have that awkward "Skip this phase on the first round" rule -- makes for a cleaner rulebook and easier to understand reference guides.

It is a bit awkward.

It was probably set as first phase to make resolution of treachery cards with "until end of the round" effects easier.

For the cost of skipping the first Mythos phase, you get the benefit of having abilities drawn at the start of the turn last until the end of the turn without awkward wording like "until the end of next round" on each and every encounter card.

It was a good choice.

I think the Mythos phase is the 1st phase so that a turn begins with problems and then deal with them. It could also come into play during missions by allowing the game to use a check such as "end the round without any xxx enemies in play." Since you don't reveal new cards after a round ends, you won't think you've succeeded and then flip another enemy.

Also lots of encounter cards say "at the end of the round test X or suffer Y consequences" if the Mythos phase was at the end of the round, they'd either have less design space or it'd have to be awkwardly worded like "at the end of the next round"

Also it'd create negative play experiences "oh great I drew this, but now I can't do anything about it for a while"