Whenever you pull the handbrake, the screen will shake and the player will wobble from side to side as the TARDIS takes flight. Flight would be more immersive with this feature (for the reason, see "Lore").
As a sort of second suggestion, occastionally, the TARDIS could use the malfunction or Stutter take off sound when taking flight, as seen in this video
"TARDIS Stutter malfunction sound FX own mix"
uploaded by "thatSFXguy"
NOT MY VIDEO
These features could be toggled the same way that the cloaking, HADS, and TARDIS Hum options are, with a button on the TARDIS monitor somewhere. There could also be a scale to set the strength of the shaking and likeliness of a stuttering take off. A similar scale to how the lighting system works, the more the %, the more intense and often it is.
Everyone knows flying a TARDIS with only one pilot makes for an extremely bumpy ride, even the Doctor can't pilot their TARDIS smoothly and they've had hundreds of years to learn.
most of us are the only pilot of our TARDIS too, so it makes sense that we'd have the effect
other than adding immersion, it doesnt really. some people may think it could be annoying, but thats why I suggested a way to turn off the feature
Not really... unless you have any questions?