I had a similar problem with my Craftsman opener from 2001 (made by Chamberlain, apparently, which is also the same company as Liftmaster). It was a gradual failure, with the remote range getting shorter and shorter until it barely worked at all, even a few inches away. You'd think this was a dead remote battery, but it happened with both remotes, and a new battery did not fix it. The wall button always worked.
I do have an LED bulb installed, but that turned out not to be the problem. It did, however, put me on the right track: I discovered that the remote problem only occurred when the light was turned on, regardless of what type of bulb was installed (incandescent, LED, or even no bulb at all!). If I waited five minutes for the light to turn off, the remote worked again.
This looked to be a controller board problem. I took the board out of the unit and inspected it closely; there were no obvious failures like burnt components or bulging/leaky capacitors, but one resistor was a little discolored. I couldn't tell what value it was supposed to be, but based on voltage readings taken up- and downstream of it, I'm pretty sure it wasn't the main problem.
The controller board was available from the appliance repair sites, but for more than I wanted to spend ($130, IIRC). For that price, I'd get something that wired into the button, either a universal remote or some Arduino thing you would wire into HomeKit. But then I checked eBay and found it a bit cheaper. More importantly though, one of the sellers offered a mail-in repair for $40, or a component kit for $6. That was well worth spending the $12 (with shipping)!
This was the seller I purchased from.
The kit came with a couple capacitors, a resistor, diodes, and the 5V regulator, and they emailed me a link to a repair guide on YouTube. I just replaced the capacitors and the resistor since the diodes and regulator tested good, and it 100% fixed the problem!
