It's like driving stick. It becomes easier and more natural over time, but you'd be a fool to stop thinking about it altogether.
I start by picking my suit (or jacket) for the day. That choice is what drives all the others. Then I pick a shirt. Between the two, I have no trouble finding a pair with enough variance in scale and which will go together color-wise. The only thing that might be lacking is contrast variance (say between a navy blue suit and a dark blue bengal stripe shirt). Still, that's no problem since i can alway "fill in the gap" with my square or tie. Thus, I could probably pick my shirt blind.
Once the shirt and suit are picked, I think about where I can add some variance in texture, contrast, color, etc., or fill in the space where I already have two extremes present. That dictates the tie or square I pick. Which I start with is a matter of whim. Doesn't really matter. Whatever one fails to do, I use the other for. Done.
By the time I've gotten to the end, there is likely enough balanced variance across the board. If not, I re-think the square or tie. I never go back and change the shirt or suit. They are never the offending agents, and are the hardest to switch. 99% of the time, if I switch out anything, it is the pocket square. The other 1% of the time, it's the tie.
The whole decision process takes maybe 15 seconds.