Not knowing your altimeter is connected to your transponder, and that you provide the altitude along with your squawk to the atc screens is a pretty glaring wtf.
I can forgive the pilots for not knowing that ATC is getting altitude information from the plane. They're not air traffic controllers after all, and the training back then was a lot less thorough than it is today.
What's more alarming is that they had a perfectly functional radar altimeter and ought to have known that this is a completely self-contained system; it doesn't depend on the flight computers, the pitot-static system, or any external information such as your current position or a programmed flight plan. If it's telling you that you're only a couple thousand feet above terrain, you really should listen to it and respond accordingly.
On general aviation planes there are usually two altimeters that are independent. The instrument a pilot uses to fly with (round dial gauge or digital version in glass cockpit) and one inside the transponder itself. If you lose your altimeter you actually can call up atc if you have a mode-c transponder and have them read it to you. It’s separate.
No idea how it works on a passenger jet, but I would be shocked if it was different.
On the aircraft in question (the 757-200) there are 3 altimeters (captain, FO, standby/backup) fed by two different colocated static ports (that were both taped over). The transponder sends the altitude reported by the captain's altimeter.
This is why the crew trusted the captain's altimeter over either of the other two because it precisely lined up with what ATC was telling them. Neither set of parties knew that they were the same incorrect figure derived from the same source. The captain correctly diagnosed that the entire pitot-static system had gone to shit, but still trusted ATC's figure, right up until they started hitting the ocean, even with the GPWS alarm from the (correctly functioning, entirely separate) radar altimeter blaring in the cockpit for over a minute.
ATC does have access to radar that locates aircraft via skin paint. TBH, I’d push that back on ATC. If I’m confused about my altitude, it’s not because I suddenly can’t read the numbers on the dial. ATC should understand that the confusion can be resolved by checking with an outside source.
Primary radar as used by ATC is extremely limited. It is not designed to identify targets at all; nor does it have particularly good vertical resolution. It can tell you where in 2D space a plane is, but without secondary radar and a cooperating transponder, it can't tell you what plane it is, and its uncertainty with respect to altitude can be several thousand feet.
Civil primary radar usually isn’t 3D and does indeed depend on transponder, i.e. secondary radar, supplied altitude data.
They should know what their radar supports in any case, and get suspicious when an aircraft asks them for a readback of airplane-supported altitude data.
What's more alarming is that they had a perfectly functional radar altimeter and ought to have known that this is a completely self-contained system; it doesn't depend on the flight computers, the pitot-static system, or any external information such as your current position or a programmed flight plan. If it's telling you that you're only a couple thousand feet above terrain, you really should listen to it and respond accordingly.
Neither of the pilots did.
EDIT: https://www.youtube.com/watch?v=jIz6vODilro