Now, this is a pretty deep concept that really made me think.
What you are essentially saying is that the analyzer is more of a "red flag" if the reading is off, rather than the primary method of measurement. The implication is that a reputable fill station has procedural and regulatory safeguards in place to ensure correct mix and that these safeguards are the primary (not secondary) source of data, with oxygen sensor confirming rather than determining the oxygen content. This also means that if there is a discrepancy, the correct action is not to use the gas until one can figure out what's going on, instead of going off of the sensor.
With regards to the multi tank dives, this means that the calculation based on residual pressures should always be performed (with dumping of the tank as a convenient shortcut,) then confirmed with the sensor. (Of course, since we are talking about recreational dives it suffices just to get to the ballpark and confirm that the ballpark is nowhere near the limits, but the principle is there: "calculate/estimate, confirm" vs. "ignore.")
What dumpsterDiver said is pretty much spot on what I typically do. Confirm your mix but you should hopefully already have a good idea what it should analyze out to if you knew your starting mix (known gas) and the pressure left in the tank.
A typical scenario for me. On Saturday I know I'm diving a specific wreck so I blend some Nitrox 36% in my tanks because I want a longer NDL without going into deco limits. Not getting into too much detail but since I am partial pressure blending, my tanks are O2 clean. I add ~661psi of pure oxygen and top off with air (20.9%) ~2839psi to have ~3500psi. If I filled slowly and accurate enough I already know my tanks should analyze out to ~36% but there of course will be slight variants on that based exact PSI, tanks cooling, etc. So let's say my tanks analyze out to 35.7%. Close enough for me to call it 36%. Even a typical +/- 1% variance doesn't bother me a lot.
Part of any standard in a Nitrox course should be to always analyze your own gas that you're diving. I don't care if the dive shop already did it for you. The key here is
trust but verify... Almost all will make an O2 analyzer readily available. If they don't I would be concerned but I happen to own one so I always check my own mix. Always analyze a gas that you're going to be diving. Do not blindly trust the label or someone else. I see a lot of experienced divers skip this part and go by what the dive shop says (if they didn't see them actually analyze it) or what's on the label. The point here is people make mistakes and since unfortunately no one is perfect you should always do your own independent verification.
So the second part of the part of the scenario. Since I personally analyzed my tanks myself I know I was diving 36%. I am diving Sunday but the depth is relatively shallow 40-50ft so I don't really care about what Nitrox mix I have. I know there is no way in hell I can exceed my MOD for that mix unless I take a shovel with me but I do want to at least know what is in my tanks.
Not a problem, I will top off with air and do some quick math. Since I am terrible at math, I use a program called MultiDeco which does the work for me but here is a simple way to do it (already explained in another post I think). There is probably a better way to represent this but since I'm lazy here is how I might do a quick spot check. Since you have physics background, I know this must be pretty basic for you but there are a lot of people diving that surprisingly don't know how to do this.
Tank 1: 800psi of Nitrox 36%
36*800psi/3500psi + 21* 2700psi/3500psi
8.22 + 16 = 24.42% when topped off with air
Tank 2: 1400psi of Nitrox 36% (perhaps I ended the dive early for some reason?)
36*1400/3500 + 21* 2100psi/3500psi
14.4 + 12.6 = 27% when topped off with air
Before I even pull out an analyzer I should at least have a ballpark idea of what is in my tanks and what they should analyze out to. If I don't it could mean a couple possible things, my analyzer is FUBAR'ed or perhaps I was not diving with a known gas to begin with (bad...)
There is a quote on here that I think is very appropriate in this situation which has to do with decompression algorithms. Nevertheless it still fits the scenario.
Rick Murchison:
Decompression algorithms are akin to measuring with a micrometer, marking with chalk and cutting with an axe.
For our purposes these numbers could be "good enough". So now I am doing two dives with 24% and 27% in my tanks. I *might* as well treat 24% like air but what about 27%? (MOD 138ft at 1.4) It really depends on the dive profile I am doing. If I am diving to 140ft then yes by all means but if I am diving to 40-50ft I really don't care too much about tracking my O2 exposure based on these mixes.
Back to your original question on O2 exposure tracking. Would these mixes be significant enough to take into account of O2 exposure (CNS and OTU exposure)? Answer: It depends.
How deep, how long and how many dives, multiple days of diving, previous mixes used, etc. I think you'll find that in most recreational dive scenarios you will not be anywhere close to these limits which partly explains why the experienced divers on the boat didn't care or perhaps it was out of ignorance.
Either way, once you know what is in your tanks you can make an educated decision on if the mix should factor into your dive planning.