If the dash display promptly shows all gear positions with no errors, I'm thinking that it is likely not the issue. Also, a visual observation of the linkage itself shows a full throw.
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Transmission pressure ports to test,
You may start with just disconnecting any of two transmission cooler hoses and checking if ATF flows. That will answer to the question: "Does fluid pump works?"
To be fair, I am very much at a point where the entire transmission (and motor) need to come out regardless of what I find, so I suppose I need to start pricing transmissions and weighing the cost vs. benefit. 200K on the car, it is due for a timing belt, I paid 2,000 for the car, and will likely never get out of it what I will wind up putting into it (I haven't even priced transmissions). Much to ponder.
Did I read correctly that the fluid level is at the hot mark when the transmission is cold? Take some fluid out so that its at the cold mark when its cold. I overfilled once and the thing barely moved. If that does get it going drive it around a bit till the trans is actually warm and then adjust accordingly.