In my experience that's where all the time saved by automating rote calculation went. Maybe this is a generational thing but for a lot of my peers Ti-84s were their first experience with programming. It's probably the only context where most students are exposed to computing that doesn't involve always connected data-harvesting.
> going to the computer lab
A student has no incentive to write programs if they aren't actually going to use them.
> Are those rare instances worth every student having such an expensive thing for several years?
I remember buying mine at a flea market for literally a quarter. TI-84s have been around for so long that they're really only expensive if you're hellbent on only buying new. Every garage sale and Goodwill in the country has at least one of these for cheap.
But "practice makes perfect" isn't that radical of an idea - you'd be hard-pressed to find a task at which someone doesn't get better with practice (barring biologically impossible ones).
EDIT: you mentioned "writing a program to automate computations" and yes I agree that that would certainly help understanding. I've not used graphing calculators all that much though, so I thought they were mostly used for plotting graphs and calculating statistical measures such as mean, standard deviation, percentiles etc. And I don't see much need for program-writing on the part of the student to do all that. The student might be far better off writing simple Python or JS programs to do those things.
1. https://link.springer.com/article/10.1007/s10649-017-9788-x