i feel like surely there must be a way to "screen-record" digital brushstrokes as they are being drawn, in a high enough quality that it can be integrated in my final project (movie will be in 4k).
Sure. On a 4k screen on na extremely powerful PC with a commercial screen capture software liek Camtasia that can drop a snapshot every few miliseconds or a dedicated 4k recorder hardware liunked via your HDMI/ Thunderbolt. Otherwise you'll have to put up with those tedious workflows indeed or simply move on to greener pastures like Krita, Moho, ToonBoom, possible Animate. That's simply the crux of all computer graphics. The more you veer away from simple lines and other stuff that can be expressed matehamtically, the more complicated things become because you are working against how the computer "thinks". That said, if you provided and actual screenshot of your art style I'm sure we could advise at least on some tricks to streamline and simplify your workflow so the project doesn't get that laborious.
You have a couple of options using a stroked path on a shape layer with trim paths added. You can apply distortion effects, grain, and edge modifying effects to make the animated stroke look more organic.
You could also use a tablet with Adobe Fresco. There are export options that include timelapse recordings of your drawings. Check it out here: Publish & Export options in Adobe Fresco
If I get some time, I'll record a simple drawing with my iPad and share it here. That is one of the easiest ways I know of to get the hand-drawn look.
Here's a fancy example I found on YouTube with a quick Duck Duck Go search: