I could *maybe* see adding some kind of annotation question type to quizzes, but I don't think adding Lockdown Browser support to assignments is very likely without a lot of other big and unlikely changes.
Assignments are, in many (or most for us) cases, something students visit once, get instructions, then go off and do work in something like word/excel/powerpoint/google docs and when they are done, come back and submit. A student may work on assignments for days or even weeks depending on the course. Lockdown browser just doesn't make sense in that kind of workflow. There are some assignment workflows where lockdown browser might make sense, but I think it would really create more issues and confusion in general.
Lockdown Browser is a bit more suited to quizzes because they are in general something students do right in Canvas, and start/complete in one sitting. Usually no external programs, usually no coming back to finish days later... That's where Lockdown Browser makes the most sense, as least in my opinion. The issue with adding annotations to quizzes is that I believe those are done through DocViewer, and usually docviewer is set up to more or less fill the entire browser, and I'm not sure it even supports being loaded twice (if you had 2 annotation questions in the same quiz). This is not insurmountable (perhaps force the one question at a time setting if annotations are used), but it would definitely add complexity to the code. When the idea conversations area reopens, you could definitely submit this as an idea to see what others and Instructure would think about it.