Yeah it does work on GSM.
But apple decided to disable it for CDMA which was yet another limitation on the device.
A bad decision
Well, Apple didn't 'disable' it. People make it sound like these handset manufacturers flip a switch and turn off simultaneous voice and data.
It's a limitation of CDMA. CDMA cannot handle simultaneous voice and data, period. The iPhone 5 only has one radio. The reason LTE works on OTHER CDMA phones for simultaneous voice and data is that they use the same CDMA chip as before, but ADD an extra LTE chip. So on those phones, you can talk via the CDMA chip, and pull data via the LTE chip.
With the single radio though, it can't connect to both CDMA and LTE at the same time. Think of it like the radio in your car. Your one radio can't play two stations at the same time, you need a second radio. CDMA only comes in mono (like AM) so you need two radios to get both sides of the signal (voice and data) otherwise you only get one at a time. GSM is like Stereo FM, you can bring in two channels (voice and data) at once on one radio. Currently, LTE is data only, so it itself can't carry both channels (voice and data) you have to simultaneously connect with a voice radio.
The reason it works on GSM, is that LTE is based on the same frequencies as GSM. So it's already tuned to the same station. So it is two separate technologies working together, but they are on the same station so it can still pull both channels. (LTE Data and GSM voice) So it can pull data, and simultaneously connect with the tower and make a voice call or send/receive a text message.
If you need simultaneous voice/data on CDMA, you need a device with two radios. Apple says they couldn't make that happen in the form factor of the iPhone, while still making it a 'world phone'. (One that can operate on either CDMA or GSM networks, which is useful for CDMA users who travel outside the US, which is pretty much the only country still using the older CDMA technology). There are some other handsets that are a bit bigger and/or lack GSM capability, so they are able to fit two radios instead of just one.
So, the bottom line is, it's carrier limitation. It's not like Apple can just 'turn it on'. CDMA won't work alongside LTE, the radio itself has to change frequencies to move on to LTE, and change again to go back to CDMA. CDMA itself is unable to handle both data and voice, it's a limitation of the technology.
Unfortunately, the issue we have in the United States is that we are sometimes the first to adopt certain technologies. Even if they are invented elsewhere, we are often the first to deploy it on a big scale (at least for now). That means that we get the kinks figured out, then everyone ELSE starts fresh with that knowledge in mind. CDMA is a result of that. CDMA does have a longer signal range, but it has certain limitations (beyond just voice and data) that make GSM a better deal, which is why the near-entire rest of the world uses it. But, CDMA made sense in the large and not-so-dense US back when cellphones weren't smartphones! The cost of switching those carriers over to GSM would be tremendous, unless something happens like with telephone lines, where the federal government forced providers to 'share' their lines so a customer could have any telephone provider, even if it wasn't the one who ran the telephone wire down their road. If that happened, Verizon and Sprint could use GSM towers already in existance. But I don't see that happening, and it's not as necessary with cell providers as it was with terrestrial land lines years ago (companies would severely gouge people who had no choice in telephone providers, because they had to go with whoever ran the wire down the pole! There was no competition)