Around 15 years ago I was an intern in the NMR department at a company. Someone higher up had heard about Maximum Entropy and suggested that we try a specific program to test if it is useful for us. I ended up trying this, and in the end simply could not get that software to work. It was buggy and almost entirely undocumented. These methods also tend to have a few parameters that you are supposed to set to reasonable values, which can be rather intimidating especially when they are unclear or even contradictory on what "reasonable values" are and you have no idea what the consequences of setting these values wrong are.

Applying FFT to NMR spectra is very easy, you get a reasonable result out of the box without changing any parameters. And the parameters that you can and should change are pretty straightforward to understand if you read up a bit on them.

You have to keep in mind that many, if not most people measuring NMR spectra are not NMR experts. They're synthetic chemists or other scientists that use NMR as a tool. So to be useful to these users the process of measuring spectra has to work mostly in an automated fashion with reasonably user-friendly software to process and handle the data.

As mentioned my orthocresol NUS is a somewhat common method today that doesn't use FFT to process NMR. The difference today compared to a decade or two ago is that in newer spectrometers and software this is already integrated and is easy to setup. You could already use this method earlier, but you had to set it up with separate tools and process as well with different software. That is a very large barrier, especially if you consider that any mistake you make is likely to waste hours to days of expensive measurement time.

Around 15 years ago I was an intern in the NMR department at a company. Someone higher up had heard about Maximum Entropy and suggested that we try a specific program to test if it is useful for us. I ended up trying this, and in the end simply could not get that software to work. It was buggy and almost entirely undocumented. These methods also tend to have a few parameters that you are supposed to set to reasonable values, which can be rather intimidating especially when they are unclear or even contradictory on what "reasonable values" are and you have no idea what the consequences of setting these values wrong are.

Applying FFT to NMR spectra is very easy, you get a reasonable result out of the box without changing any parameters. And the parameters that you can and should change are pretty straightforward to understand if you read up a bit on them.

You have to keep in mind that many, if not most people measuring NMR spectra are not NMR experts. They're synthetic chemists or other scientists that use NMR as a tool. So to be useful to these users the process of measuring spectra has to work mostly in an automated fashion with reasonably user-friendly software to process and handle the data.

As mentioned in @orthocresol's answer NUS is a somewhat common method today that doesn't use FFT to process NMR. The difference today compared to a decade or two ago is that in newer spectrometers and software this is already integrated and is easy to setup. You could already use this method earlier, but you had to set it up with separate tools and process as well with different software. That is a very large barrier, especially if you consider that any mistake you make is likely to waste hours to days of expensive measurement time.