But considering that Darth Vader was a Sith Lord, and a Jedi before that, are we entirely sure he wasn't secretly controlling Tarkin through a
Jedi Sith mind trick?
It doesn’t seem likely
First, there’s the obvious matter of Tarkin disagreeing with Vader in A New Hope:
TARKIN: Obi-Wan Kenobi! What makes you think so?
VADER: A tremor in the Force. The last time I felt it was in the presence of my old master.
TARKIN Surely he must be dead by now.
VADER Don’t underestimate the power of the Force.
TARKIN The Jedi are extinct, their fire has gone out of the universe. You, my friend, are all that’s left of their religion.
This isn’t the only instance, of course. He orders Vader around, suggests alternative interrogation techniques, and otherwise doesn’t act at all like someone under the influence of a mind trick.
Then, too, mind tricks are not effective on the strong-willed. While it’s always been a little unclear what that means, it seems likely that Tarkin is strong-willed: he’s a leader, not a follower; intelligent and driven.
They had already witnessed how well he could fight, scale mountains, pilot a gunboat, and succeed on a sports field, and—as crises like the one at Halcyon grew more common—they grew to realize that he had a mind for tactics, as well; more important, that Tarkin was a born leader, an inspiration for others to overcome their fears and to surpass their own expectations.
Perhaps most important, the Emperor valued Tarkin. It seems unlikely that he would have allowed Vader to control him. More than that, he wanted Vader and Tarkin to work together as equals:
During the Clone Wars, Sidious had made every attempt to promote a rapport between Skywalker and Tarkin, but the relationship had never prospered to his satisfaction. Then came that business with Skywalker’s Togruta apprentice, Ahsoka Tano, which, while it had provoked further disaffection in Skywalker, had also created a rift between him and Tarkin that perhaps had yet to mend. Yes, they had partnered since the end of the war, but—to Sidious’s own annoyance—absent a true appreciation for each other’s talents.
If the Emperor wanted Vader to appreciate and take advantage of Tarkin’s talents, he wouldn’t have looked too kindly on Vader using mind tricks on him. Those influenced by mind tricks tend to be rather pliable, and not necessarily the best military tacticians. And that’s what Sidious wanted, not a puppet.