If what you show in photo #3 is what came out of the current pot intact, then yes, it is severely pot bound. And contrary to what you may have read, no plant prefers to be pot bound. You would not find that happening under natural growing conditions so why should it be any different as a potted houseplant?

It needs a slightly larger pot with a fast draining soil mix, something you would use for cacti and succulents. I grow almost exclusively in containers, indoors and out, and I rarely pay a lot of attention to the time of year when repotting, except for very temperate climate plants that go through a full dormancy - Beaucarnea do not.

The root congestion and lack of any visible potting soil would indicate that a) the plant may be drying out too quickly/a need to bump up the watering frequency. Drought tolerant or not, plants need to be able to access moisture as their biology demands and there is nothing visible to support any degree of soil moisture under current growing conditions. And b) there is nothing to provide any nutrient supplementation. Plants grown in containers with a soil-less mix will require you to provide nutrients on a pretty regular schedule regardless of time of year.

I seriously doubt it is any sort of disease and I see no signs of spider mites (stippling and webbing). It just needs repotting. And some fertilizer.

Gotcha, thank you so much for the help! I'll get on repotting it. You even cleared by concern about repotting it during this time of year.