Your question is a good one, and it's one that I have found myself pondering the last few days. I did my iron fitting and moved to graphite shafts in a TX flex with weaker lofts and one degree upright for the first time and I've asked myself, "Will I see the same results when I put the new irons in play?". I feel that if you have gone a reputable fitter then we have to trust the process. My thought is the test-retest reliability would produce similar or consistent results, looking at it from a macro level on what the fitting is designed to achieve, loft, lie, feel, shaft flex, it creates a club(s) that puts you in the best position to achieve repeatable results.

As amateurs, I think we always bring a different swing each round, how different is dependent on the individual. The biggest X-Factor you highlight is you as the player, even the pros will have "off days" when it comes to their swing, albeit probably less frequently than you and I. It's a reminder of how important it is that when we go to fittings that we try our best to bring our "swing", fittings can be nervy we want to do well but adrenaline and overthinking can creep in, at least for me.

If you have different swings, then maybe you're not the best candidate for a fitting. Once you establish a repeatable swing, then a true fitting matches the clubs to your swing.
Look at it from the other side. You said you warm up to decide which swing works that day, so should you buy several different sets of irons, take them all to your warm up, and then decide which set you're going to play that day?
The answer is yes a one day qualified fitting will produce repeatable results, if you can produce a repeatable swing. The clubs are the clubs. They don't change to match whatever swing you are playing.

Part of the fitting with a good fitter is a discussion during the process about whether you are hitting with your normal swing or if you are a bit off that day. They are interested in hearing what your normal shot shape is and your normal miss. Even when off, your basic swing speed will not vary as much so that part will be pretty consistent. If you feel like you are way off during your fitting, just reschedule and do it again be for ordering clubs to see if you get the same results or if it changes a bit.

Since I get to warm up with my current iron(s), what I’m doing with them will let me see what the fitter is doing with new options. I might have to hit 6-7 shots on a bad day to get 3 unshackled/thinned shots, but we go from there.

If there is not enough change on launch, carry, or descent, dispersion isn’t improved and the feel is similar, then a new set comes down to need vs. want. Gen 3 & 4 have improved sound and feel that I justified new clubs. Subjectivity is an important part of the process in going to just the next generation.