Inertial systems integrate all measured accelerations over time to compute present velocity and position. They are periodically corrected with external references. The accuracy is limited to the accuracy of the accelerometers, the sampling frequency and the accuracy of the digital math, among other things.
This IMU idea could be used, but I am not aware of any current commercial IMUs in airliners - but I am not an airline pilot so that does not count for much.
If I were given conflicting information I would see how the plane is acting. I have flown planes with various primary instrument failures over my decades, some in actual instrument conditions, and always had sufficient secondary sources of data to determine the situation. As an example in a piston plane: Tachometer fails? Cross check fuel flow, manifold pressure and airspeed - subject to mixture control which can be reasonably set using "lean to roughness and enrich" method. Been there, done that. I have had an altimeter fail insidiously slowly, when water got into the static lines (despite redundant ports and properly routing them upward), while in actual near minimums in a non precision approach. (I used GPS altitude cross reference and odd seeming behavior of altimeter needle to diagnose and switch to alternate static.)
Importantly, if the plane is flying straight and level and sufficiently fast, over time this becomes a "feel" and you can tell if something is extraordinary. Maybe not on airliners, but definitely the piston planes I fly a lot and probably the small jets I fly on occasion.
This is why some of these things are mystifying to me. If your engines are set to a certain power and seem to be operating properly, and your attitude is just so, in the absence of unusual meteorological conditions your plane should be performing in a manner consistent to your last experience. If your stall warning is going off, well, maybe the instrument broke. Air France is really scary mystifying.
Look, I am a software engineer so I get it, people can get locked into a misconception about a fault and not step away long/far enough to consider other alternatives. It happens a lot in debugging. But for whatever reason my brain works much faster and more dynamically when shit happens in the left seat of an airplane having problems.
Maybe you didn't intend it but this sounds a lot like "couldn't happen to me!"
I meant to mean "It has most definitely happened to me" although not yet in the context of piloting - but in other contexts.
For the Lion flight, winds were light and the aircraft was at low altitude, so ground-speed would have been very close to airspeed.
In any case (as pointed out by earlier comments) a proficient pilot should have no trouble flying in daylight and good conditions using only the "Mark I eyeball". It appears the aircraft crashed because the crew didn't fly it - a surprisingly common occurrence for automation-dependent crews!
I still think I'll be surprised if accelerometers provide useful velocity information over timescales useful to a pilot. But maybe that's because I only have experience with cheap MEMS accelerometers from around 15 years ago.
More useful at the time would be to set a "sensible" power level and maintain approximate altitude ... and looking out the window is usually helpful!
An inertial measurement system cannot measure velocity. It's insensitive to which inertial frame it is in. (to move from a non moving frame to a moving frame, you must accelerate, which it could detect. but then you have to integrate these accelerations over time)