My suggestion is that every bit of energy goes into Dragon 24/25 and Spider 25 basically for weeks or more.
Lots of folks even in my own clans just log in and spend a little bit just doing dailies and maybe working on leveling a bit of food. They may run 45 Dragon runs overnight sometimes, or even most of the time, and they wonder why they are being left behind.
You said you don't want to think about a go-second team, so realistically you have to run Dragon 20+ for weeks, with all your energy and maybe all your gems for energy, before you'll see speed improvements. No leveling food champs unless you're solo farming dragon. No going hard on some random Ice Golemn tourney, or running IG at all. Or anything else that won't help your speed problems.
And, if you don't think this is a process that will take a full month or more, you have to adjust your expectations. This will be a long process. Fortunately, people that already have a 340-360 Arbiter and a 320 high accuracy Warlord, they have to run exponentially more dungeons than you just to find a piece that improves their account even in the slightest, not even just with speed. So you can definitely close the gap, but only by being hyper focused.
That's my best advice at least.
While I appreciate the advice, and the honesty, I have to point out that this is an outrageous take on the game.
There is NO way I'm going to be "hyper-focused" on anthing in raid for the next month. LOL.
This is not a rude comment at all, just an honest one.
This idea is exactly the idea that will eventually sink this game. The time requirement is just getting silly at this point. I've never been "hyper-focused" as a Raid player, and I never will be. That's just the facts of the case.
Again, I really like the honest advice, but I think the game (and maybe the community) are the ones that need to adjust their expectations of players...not the other way around.