To be honest, if you never checked TA it would probably not be the end of the world. It is always the last thing we say to worry about.
Now, with that being said if you have what we call a quality test kit (Taylor K-2006C or a TF-100) at the beginning of the season run all of the tests to get a baseline. Then, worry about what is really important FC & pH.
You are a hop skip & a jump from me, so here is what I do- - -
I keep the pool open all year
During the winter I check FC every couple of weeks.
In the spring as it starts to warm I do a full battery of tests FC, CC, pH, TA, CYA & CH I adjust CYA & CH at that point for my plaster pool. You have vinyl, so unless it's outrageously high CH means nothing to you.
I have a Stenner Pump auto-chlorinating my pool so every 2 or 3 days I check FC & pH. Once every 5 or six weeks I test TA, CYA & CH just to see what is going on.
Generally, during the year my pool gets chlorine and acid to bring the pH down. That's it.
Pool stores want to make is seem like your pool is going to fall apart if you don't constantly adjust every parameter. Why, to sell you stuff to adjust every parameter. Oh, your TA is low, add this. Oh, your pH is rising too fast, add that. Your vinyl will crack if you don't add this, something else will happen if you don't add that.
There is no compelling reason whatsoever to check TA first. If anything, you should run pH and Chlorine before any other testing, because they are the ones which are time sensitive. If you grab your sample, then run your tests immediately (and you should) TA should not be a priority for first in line. It and CH can wait, and fight it out for last place in line. If you are adjusting pH with Pool Math, you need to know the TA number. While TA is important for various reasons it's not something you should put laser focus on or worry over too much. I'm not saying ignore it, but focus on controlling pH, and Chlorine first.
I agree and the FC level is probably first of all because it is the only thing to KILL all the stuff that can at best turn your pool green and smelly and at worse kill you. pH is a very close second. This is why they're side by side on the OTO test kit which is all we used back in the day along with stockpiles of Sodium Hydrochlorite and Muriatic Acid always on hand.