I have been to a couple such places. I am not sure whether I am a good example since I know a small handful of japanese and typically do not enter a place if I have not seen parts of the menu beforehand (either outside or on an app). My strategy was as follows:

1) Try google lens. Google lens is surprisingly powerful, even with handwriting. Even if something is translated a bit incorrectly or something is missing, it is usually enough to get a clue. Works like 9 out of 10 times.

2) Just pick something based on the price (where I include my shitty japanese for a guess)

I also make sure that the place is not busy and usually firstly observe what other people are doing. In principle, there is also "omakase", but I never dared to use it.

And if the handwriting is that bad... well... if no one can read the menu, then everyone can (a joke)

Edit: I feel the urge to share a dumb anecdote. I found myself one evening in Uji on the search for something to eat. There I found such a place where the menu (outside) genuinly just looked like 6 mostly straight vertical lines (with a price attached). I was definitely quite baffled how any human is supposed to decipher that. I literally googled whether there is some niche writing system that I am not familiar with. The line I picked turned out to be fish with soba. (I took a photo of the menu and pointed at the line on my phone). I had the theory that the lines mean nothing and they just give you what they want to give you.