Why must Vr be set so precisely?
This answer is written for air transport category aircraft.
Introduction
During take-off there are three operationally significant speeds that ensure a safe take-off:

V1 - the take-off decision speed
VR - the rotation speed
V2 - the take-off safety speed

In addition there are three technically important speeds:

VMU the minimum unstick speed
VMCG the minimum controllable speed on the ground in case of an engine failure
VMCA the minimum controllable speed airborne in case of an engine failure

V1 is the take-off decision speed. Below V1, take-off can aborted safely without overrunning the end of the runway. Above V1 the aircraft is unlikely to be able to stop before the end of the runway so in case of an engine failure it is safer to get airborne. V1 needs to exceed VMCG to ensure the aircraft doesn't veer off the runway when accelerating to take off with a single engine.
VR is the rotation speed. At this speed the aircraft should be rotated and will get airborne soon after. The rotation speed ensures that, in the case of an engine failure, lift-off is possible and
V2 is reached at 35 feet at the latest. At the end of the rotation manoeuvre, the aircraft should be airborne. Otherwise the tail will hit the ground. Therefore, VR will need to exceed VMU.
V2 is the minimum speed that needs to be maintained up to acceleration altitude, in the event of an engine failure after V1. Flight at V2 ensures that the minimum required climb gradient is achieved, and that the aircraft is controllable. V2 speed is always greater than VMCA to ensure the aircraft is controllable in flight if an engine failure should happen.

Answer
What will happen if we just pick a VR that &quot;feels good&quot;?
If VRfeel is lower than VMU, we will end up with a tail strike. Even if the rotation would stop just before the tail strike, the increase in aerodynamic drag (lift is generated) will cause the take-off run to be longer than needed, risking not being able to climb over objects at the end of the runway. In addition, if the aircraft has an engine failure during rotation, it may not be able to reach V2 and could become uncontrollable and crash.
When VRfeel is too high, the aircraft risks exceeding the maximum safe tire speed. There is also a risk of not reaching VRfeel before the end of the runway.

References
Airbus operations briefing note on takeoff and departure operations; understanding takeoff speeds
Boeing article: exceeding tire speed
rating during takeoff