Why does V1 increase with weight?
Other than being controlled by Take-Off Run/Distance Available (TORA/TODA) or Accelerate Stop Distance Available (ASDA), all field limits, maximum $V_1$ is also controlled by $V_R$ (Speed for rotation) and $V_{MBE}$ (Maximum brake energy speed, but let's ignore $V_{MBE}$ in this context). $V_1$ must not exceed either.

$V_R$ increases with increasing gross weight.

This means that at low gross weight, theoretically, if the aircraft had not yet been rotated past a $V_1$ limited by $V_R$, sufficient distance may still exist to stop the aircraft on the remaining ASDA.

At higher gross weights, the higher $V_R$ required might allow for a higher $V_1$. Therefore, in response to: "So why isn't V1 reduced to allow for better stopping?": The higher $V_1$ still provides adequate margin for stopping on the ASDA, but gives you the option to abort the take-off at a higher speed with the increased $V_1$.

Rejected take-off (RTO) is still initiated when the engine failure is recognized, so a failure before the "increased" $V_1$ will result in RTO at the time of recognition of the failure, and not at $V_1$.

If performance allows, I prefer $V_1$ = $V_R$, as opposed to say $V_1$ = $V_R$ - 20 kt, on a short, slippery runway, which would imply that given an engine failure past $V_1$ you'd have to keep accelerating the aircraft with asymmetric thrust, on said runway, close to $V_{MCG}$ (minimum control speed on ground, which can be the lower limit for $V_1$).