Do coffee and tea need to be agitated while brewing?

Tea and Coffee contain a spectrum of flavour compounds and most certainly, not all are pleasant.

If you've ever had an over steeped tea or over brewed coffee you'll know that the optimum extraction point is long before maximum extraction. Where you are no longer extracting just pleasant compounds and picking up higher concentrations of things like caffeine, bitters, etc.

Agitating tea or coffee runs the risk of (or facilitate) reducing the time it takes to extract the unpleasant part. On that front, experiment with not yo-yo dunking tea bags and squeezing the tea-bag with the back of the spoon (mild forms of agitation) to see the taste difference.

There are many different methods of brewing tea and coffee, so lets consider three just as examples.

Before looking at them, however, it is worth noting that water is composed of tiny molecules that are polar. This makes it an exceptional solvent for almost all polar molecules, which is the reason why it has the reputation in chemistry as the "universal solvent" which is nearly true.

The flavor and aroma molecules in tea and coffee tend to be larger molecules that are dissolved into the water.

Tea Bags

Obviously the tea cannot be agitated. However, the volume of water is large, and it is cooling so there is going to be some convection.

The tea bags are also designed to have a large surface area exposed to the water. That is why they are thin, instead of compact and spherical.

The extraction is good enough.

Drip coffee

The water drips past the grounds. Even if a given volume of water, as it passes through the grounds were to become saturated, the next volume of water that passes it may not yet be saturated.

The time, temperature, and drip rate are calibrated (via empirical experience) to optimize the extraction.

The motion of the water under gravity, dripping down, ensures that no single volume of water remains exclusively in the vicinity of a particular section of grounds for a long time.

French Press Coffee

This is the only common coffee method where there is little natural convection, nor any movement of the water due to pressure, gravity, or other means.

It is also the only method where stirring the water and grounds once or twice is recommended.

However, even so it is not essential, as all of the grounds will be surrounded by some volume of water which extract the flavorants locally. When the plunger is used, the various regions will be mixed.

Conclusion

In general, the extraction power of the water, especially at brewing temperatures, is sufficient to the task, even with little additional agitation.

With tea, there is no need to agitate the pot/mixture in order to improve the extraction. In fact, you need to stop the extraction before the unpleasant components get too strong.

Since you asked for a physical answer, I think it's the Brownian motion from the boiling water that takes care of everything not done by the fluid dynamics of convection.

I'm somewhat of an expert in tea, though not in physics, mind you, so you can take the answer with a grain of salt.