This is a topic that's personal to me, although I think it belongs to a broader discussion about what baseline competencies are necessary in the field.
UX Design is one of the few tech disciplines that places a high value on the handmade and handwritten. The expediency of hand-crafted work lends itself to the rapid iterating process most UX designers are involved in. Work that has the mark of human imperfection is also more relatable to those viewing it, which helps in conveying complex ideas to the team. But as anyone with bad handwriting can tell you, there is an "easy on the eyes" version of human imperfection that is prevalent in the field, and it's not easy to replicate it when your handwriting falls beneath a certain threshold.
As long as your handwriting can be read, this shouldn't matter. All of a UX designers hand-crafted work is meant to be disposable, they do not require precise spokes for other team members to fit their rods. All of the work that does require that level of precision is digital.
But there are still some negative ramifications for a UX designer with bad handwriting. In studies, it has been shown that people associate bad handwriting with stupidity. As someone who works under the label 'designer', bad handwriting is also likely to signify a lack of artistry and visual sense.
So as I continue as a UX designer, I will be paying close attention to how my sloppy hand-writing is received. Does it serve as an ice-breaker that loosens people up and gives them space to create? Does it undermine peoples confidence in my capacity? A little of both? We shall see...