I think this makes assumptions about the speed at which things are currently moving.

If you had reason to believe that the client was really, truly, about to pay you, then you wouldn't have written the blog post.

If you get legal involved, the speed at which you move towards getting paid may change to become nonzero. At some point the speed of the client's legal team is irrelevant, because a sufficiently slow legal response by the client translates to a court judgement against them.

It is impossible for things to get slower than nothing happening. There’s a very good chance that they’ve decided to just not pay you until you show you’re willing to get lawyers involved. That is a fairly routine thing that large companies do with smaller suppliers.

Your problem is that nobody cares enough to deal with paying you because maybe someone forgot a purchase order or something. Your solution is to make someone care. Sending a letter to their legal team will make someone care (and they will in turn kick the accounts payable team into action). They will receive letters like this constantly. Your situation is not unique, except for the fact that you’re not taking action.

I've been in this situation, a little bit of a legal threat does wonders

This is likely incompetence, not malice