Specifically on the Escalation Die, it seems you haven't looked past just one aspect of it. It's not only a flat additional +1 to your attack bonus each round to make the fight go quicker, there is more to it than that.
Yes, it does get introduced in round 2 and goes up by one each round from there. This is meant to add a feel to the combat, because the monsters will enter the fight more powerful than you are. It's meant to evoke the feeling of a struggle when players initially encounter the monsters, it's challenging, the outcome seems bleak, but soon enough, the tables turn in the players favour and they start to gain the upper hand in the fight and leave triumphant. Using the term from the book, it's meant to represent increasing momentum. Because of this, if the players aren't pushing this feel, the increasing of the Escalation Die can cease until they do, or it can be reset to zero if combat has all but stopped. This shows a lot more thought, effort and the use of a mechanic to better serve the flow and feel of the game. It's certainly a lot more than "to fix the problem of longer battles".
There are other things that use the Escalation Die in the game. It can be used as a reference for damage on a miss, when players can use some abilities. Enchantments can use the Escalation Die too. It can tell the GM when a monster can gain an attack (so you might want to kill it quick before the big hurt comes), how many attacks it can use, or what type of attack it will be. Interaction with certain monsters can nullify the Escalation Die, or inhibit its advancement, as could some ongoing effects. Some monsters get the Escalation Die as an attack bonus too, since they're badass monsters and need to continue to be that way.
As you can see, it's much more than just a superfluous addition to the game.
If you're having a hard time imagining how this game performs at the table, check out some actual plays, there's a bunch of them out there. It's a lot more elegant than you're envisioning it, especially with how characters have a relationship with Icons and how that can influence the game, that the combat system is quite well thought out, the One Unique Thing of course, and more. It's a great step forward for the d20 style of game.