I was waiting to see if anyone with actual facts to offer would answer this, but since not I'll offer an opinion that may be a little light on facts. As in the US, in Mexico vehicle registration and driver licensing is the responsibility of the individual Mexican states and requirements may vary from state to state (Baja has two, Sur and norte). That said, I don't believe the problem with not having a Mexican license is that you can't actually buy a car but rather, if there is one, is either with registering the car or buying the insurance you'll need to register the car. For a data point I know my brother-in-law got interested in a driver's license only after he bought a car in Sonora and began to deal with registering it in Guerrero, so that may have been a requirement there.
To start with the last question, it is now possible to import certain US vehicles into Mexico but only if the car is a NAFTA vehicle between 5 and 10 years old. There'll be taxes and brokerage fees to pay. It isn't quite clear to me what this fixes, however, since the imported car still needs to be registered in your state of residence which will have the same requirements for it as a car bought in Mexico. If the idea is to register the car in the US, use it that way now and import it later, you'll need to check the legality of this if you don't have a US license since there is a Mexican federal law restricting who can drive foreign-registered cars in Mexico. For a US-registered car I know US license holders who aren't permanent residents can while Mexican license holders can't, but I have no idea which group you, as a third-country licensee, would be in. Worse, you'll now be looking to buy two insurance policies for the car without an appropriate license, one for the US registration and an additional one to cover your use in Mexico. I'm pretty sure you can get the former without a US license (for a price), but I'm not sure about the latter.
Given this, plus the fact that you are a Mexican resident entitled to apply for a license, I'm wondering why the solution to the problem of no Mexican driver's license isn't instead to just get a Mexican driver's license? I'm pretty sure that when you step off the plane in Mexico as a resident you'll have just about all the documents you need to apply for one (know your blood type and bring a birth certificate!) save something showing your residential address, and opening a bank account with your address(-of-convenience) might provide that. If your Spanish is bad find a translator to help fill out the application form and go to the office with you. Spanish is not required to drive. It is possible they won't require tests, but if you need to do a written knowledge test the state may very well have an English version (maybe no English study material, but if there's only one English test a couple of tries might get you past that) or, failing that, see if they'll accommodate you by letting your translator help or, failing that, ask if they'll waive the test because they don't have one in English or, failing that, see if you can find and memorize the question pool in Spanish or, failing that, make your first Spanish lessons be about understanding enough of the test to pass or, failing that, maybe you can find someone (e.g. the dealer wanting to sell you a car?) who knows someone who can "help" with the application (you can tell I lived in Guerrero...). If they also want a road test, just do it. If you treat getting the license as something you just have to get done now by any means necessary I suspect you'll find it doesn't require months of study, only some perseverance early on.
I personally think you are much better off buying the car in Mexico in any case, and if that really does require you to get a license first you should just deal with that. Make sure the insurance policy you get covers you for short trips to the US and you'll be fine for that too.