Is plane of symmetry a necessary and sufficient condition for optical inactivity in coordination compounds?
The number $\pi$ for bowhead whales† is still 3.1415... Same thing here.
Chirality of coordination compounds is the same thing as chirality of organic compounds. If all your symmetry elements are just rotations, you have it; if some are improper rotations, then you don't. It is as simple as that, and is not going to be any simpler.
Can you just check for the planes and call it a day? Well, you kinda can, and that will do, until you come across a compound which doesn't have a plane, but still is achiral because of some other symmetry element.
Are there such compounds? Well, that's like asking whether there is a place on Earth where three bricks stand on top of each other. You may google long and hard, or you may apply for a travel grant to visit countries with rich traditions of brickwork masonry... or you may just take three bricks and put them on top of each other. That's it; problem solved.
Here are my bricks. I claim that the tetrahedral $\ce{[Zn(NH3)4}]^{2+}$ has an inversion axis $\bar4$ and hence is achiral. You may point out that it has a plane as well. Fine. Let's give it some substituents: change each ligand to $\ce{N(CH3)3}$. What? Does it still have a plane, given that everything which rotates is rotated to the appropriate position? Fine. Let's give it different substituents. Change each ligand to $\ce{CH3NHC2H5}$. Where is the plane now?
So it goes.

† There was an old anecdote about an even more old paper on biophysics which was said to include some volume calculations and some crude approximations of a whale as a cylinder, then of course the number $\pi$ showed up, followed by the hilarious phrase along the lines of &quot;The number $\pi$ for bowhead whales is about 3.14&quot;.