If there is disagreement, it obviously doesn't matter which one is wrong to detect there is a malfunction. This case would warrant the disconnection of any automatic system that could rely on or even be influenced by those sensors, and issue a warning. First thing that happens is usually the disconnection of the autopilot, and I'm pretty sure that happens when there is a faulty pitot, at least on Airbus planes. This is the first thing that happened on the AF447 for instance, AFAIK. Of course, this could be different in this case. Maybe.
In the AF447 scenario, the computer decided it did not have reliable air data, so it not only disconnected the autopilot, it dropped into alternate law. Basically, the computer just punted and said "your airplane!" This might seem like reasonable behavior, but one might imagine that handing a degraded airplane to a pilot at the very worst moment is not a satisfactory failure mode, and in the AF447 scenario the outcome was obviously bad.
ISTM that Airbus have made fairly clear and consistent design choices. The flight envelope protection will not allow the pilots to stall or overspeed the aircraft unless the sensors start playing up (in a detectable way) or the pilots deliberately disable this system (which takes a single action). Not everyone likes the choices but they are reasonable. I agree with you that it is difficult to make changes that are unambiguously better.
Judging by the AD, the accretion of changes to the 737 has produced a system that is:
A - Badly documented (supposedly a deliberate choice to save time and money during training).
B - Does not completely handover control when the sensors disagree (even warning the pilots that the sensors are playing up is an optional extra! [1]).
C - Requires two actions, both disabling the autopilot and cutting out the trimming system.
I think Boeing have been caught out by the compromises involve in updating a very old design[2], marketing pressures[3] and - possibly - by their own propaganda regarding how traditional the flight controls on a Boeing are. My predictions are that:
A - Training will now heavily cover the auto trim systems.
B - The sensor disagreement warning will not be optional in practice (everyone will buy it)
C - Boeing may have to change the MCAS system by adding a cut-out when the sensors disagree (should have done that from the start IMHO).
I'm surprised at what is hidden from pilots both in the manuals and by making the sensor warnings optional. If pilots get a stall warning without an explicit indication that the sensors disagree[4] they may well believe it. Particularly if they have already been surprised by changes in the aircraft's motion (driven by the automated system's intervention) and so lost their 'feel' for whether the aircraft might plausibly be close to stalling. A curious and sad inversion of the AF447 crash if so.
All of this is my ill-informed opinion. Both Airbii and Boeings are flown very safely by sensible airlines. The Lion Air crash may well be something completely different, etc.
[1] Maybe Boeing could consider in-
appflight purchases in this case? "We've noticed that your subscription does not include information helpful to your continued survival. Would you like to upgrade? (T&C apply. Always make sure you have the permission of the credit card holder. Offer not valid if you're going to sue us.)"
[2] First flight in 1967, 51 years ago and 64 years after the Wright brothers. Dates from wonkypedia, maths from my sligtly tired brain.
[3] Honestly, I get infuriated by manufacturers making an oscilloscope's actual capabilities an optional (i.e. expensive) upgrade and now Boeing's marketing department gets in on the game? And then nobbles the flight manual/ conversion course to boot?
[4] Do any of the checklists (memory or written) call for comparing the instruments?