Stirring Tea in a Teapot

I saw an episode of the David Suchet series of Hercule Poirot mysteries (Dumb Witness, an early one, rather long but satisfying enough). In this program there was one scene where the potty old lady sisters next door to the murder scene served tea to several people, and one of the sisters stirred the teapot after the water was poured, with a presumably silver spoon. Although everyone saw her do it, no-one remarked on it, and it was not relevant to the narrative in any way.

I have always read that one is not supposed to stir the teapot, but never why not.

I ask this because at work I am not allowed to have an electric tea kettle, so I am obliged to get my (not hot enough) water in a trickle from the bottled water server. At home when I pour in boiling water that action stirs the tea quite adequately. But when the water comes in a trickle not even all the tea gets wet for a while, until it gradually sinks down, so I give it a good stir. I’ve done this in spite of what I thought was standard practice for loose tea. Now I am wondering if I was mis-informed.

tl;dr version: is it desirable, acceptable, or a bad idea to stir the tea after the water has been added?

eta: I am American, so I was raised in ignorance where proper tea-making is concerned.