Can students in a specs grading setting do work without help? Do we care?
You make good points that Modern Algebra students shouldn't necessarily be able to write profs with no help whatsoever. Maybe this is a better way to rephrase the question: "How do we know that the student could make a useful mathematical contribution to a team working on modern algebra problems?" That wouldn't necessarily need to be a "one-and-done" timed assessment. It also wouldn't necessary to have an expectation of perfection. It would just need to provide evidence that the student has a reasonable understanding of the principles of Modern Algebra.
Interesting post, as always. You seemed to be arguing hard against the idea of timed assessments, but then you admitted that you still use them, so I guess you were just making the strongest argument possible for the sake of airing the key issues. I do want to push back on some of those arguments, though. Yes, timed assessments cause anxiety, but so does almost any kind of assessment. I think your argument would have to be that the amount of increase in student anxiety is not worth the additional information gained by the instructor, or something like that. Regarding the assumption that time assessments test low-level cognition, that can easily be remedied by letting students use notes (as you said) and/or by giving students clear advance indications of which facts they'll need to know, so that the focus of the test becomes one of applying those facts well. The latter can be done via various study-guide options such as via Ben Wiggins' Public Exams or my own Test Question Templates (TQTs). Both of these originated in biology but should also work for other STEM subjects, perhaps even math. (Their creators would like to think so, anyway!)