Though I’ve become very good at the basic set, the group I occasionally play with still gets easily bogged down with rules and recordkeeping. One simple change we have made that speeds things up is keeping a steady, unchanging party turn order. We make one person the full time party leader / scapegoat / hero / first player. Keeping the same order throughout really helps newer people focus on other aspects of the game.
We also do Arkham encounters and Other World encounters at the same time. We do them in party order. We go around the investigators in order. If they are in an Arkham location they get an Arkham encounter, if in the streets, no encounter and if in another world, an otherworld encounter.
So far I have found no real problem with the way they work out in games. I’m sure there may be some specific circumstances where they may mess things up, but we have not encountered it.
People still get confused with the changing turn order. It seems like its a long time between turns after you pass the First Player token. We had a player last week who ALWAYS wanted to read another Mythos phase when she was pass the First Player token was passed to her, even though one had just been read.
(Of course, this will be even more confusing that your house rule, since this player will often be moving out of order.)