Good - this is the type of analyzer I hope they would use. If they connected it to the main panel and they saw a fluctuation when a major appliance turned on/off, I'd start looking at the magnitude of that fluctuation. It could very well be that you have smaller-than-usual service conductors combined with a longer path to the transformer combined with a failing start capacitor on one of your heat-pump units (that causes the instantaneous load to shoot up to 150A or even 200A for more than a second or so). Even if you knew the fluctuation parameters that the Tesla used, it wouldn't solve your problem. Yes, it would able you to argue with Tesla the thresholds, but at the end of the day, your home electrical system is seeing a voltage fluctuation that is outside the "norms" that Tesla has observed. You need to fix those fluctuations if you want 100% charging current. If all you want to do is argue with Tesla, then I suppose you could continue banging the drum about knowing the specific thresholds.
This combination of factors is why each case can be different, and why you need to rule out various cases. It could be a single motor appliance that has a bad start capacitor or bearings that are going bad; it could be that your service conductors are horribly undersized, like mine were before the upgrade. It could be a combination of them - let's say for a second that my home was 300 ft from my transformer (0.6 kft round trip), that I had 2/0 cable feeding it (0.128 Ohms/kft), and that my heat pump had a 200A start-up load for 1.5 seconds (because it had a capacitor that was starting to go bad). My voltage drop would be V=IR, so V = 200A * (.6 * .128) or 15.36V during that 1.5 seconds. If, during charging, the Tesla saw my voltage drop by 15.5 volts (what we typically saw as "dimming lights" with incandescents) for one and a half seconds, I'd hope it would back off for the sake of safety.
It's not as simple as "handling 40 amps of heat". Heat is resistance times the square of current. If you have a high resistance (due to a loose connection somewhere), the heat generated will be tremendous and can start a fire. Yes, a smaller cable will be warmer - and Tesla uses 2 parallel, smaller conductors from the plug to the UMC and to the car. That will generate some heat, but it's nothing like the heat generated from a loose connection.
Yes, voltage fluctuations will always happen, but it's not as simple as knowing what the thresholds are. It's not like you will tune some appliances to remove just enough to keep it from happening. I have found that Tesla's thresholds are pretty reasonable - maybe they're on the conservative side, but I'd rather they protect my home than be reckless about it. It's obvious that you have something that is triggering a fluctuation (or a loose connection that makes it non-deterministic), so do some troubleshooting to narrow it down then make some decisions. Turn off all other loads and see if the Tesla charging rate stabilizes. If it doesn't, then you'll have to look at your service conductors and/or transformer loading, and you may have to argue with the power company. If it does, then start introducing loads to see what causes the Tesla to back off. Start with biggest loads first... the first time you see it back off, isolate that load for a while and see if the rest of your loads allow it to be steady. Then you know where to start looking - whether it's an aggregate problem (service conductors) or an appliance problem.
Bottom line: even if you knew the thresholds, you'd still have to perform troubleshooting to find the cause of the fluctuation.