If a pressure gradient force poofed and appeared out of nothing, then it would be as you suggest:
- The wind would start out moving from high to low.
- The wind would continue to accelerate due to PGF, and the wind would shift direction (to the right in the NH) due to Coriolis.
- At each point during the turn, more of the Coriolis turn (to the
right in the NH) would be becoming in opposition to the PGF rather
than perpendicular to it... so gradually the speed acceleration
decreases (more of the forces are perpendicular to motion rather than
in line with the motion)... and the directional acceleration (after
initially increasing due to increasing velocity $\implies$ increasing
Coriolis) also gradually decreases (as the wind turns [more south in
your picture], the PGF starts to counteract more of the coriolis).
In reality, at frictionless levels like 500 mb, because pressure development is generally gradual, the wind steadily adjusts as the changes gradually occur, and so the result is that winds are basically geostrophic always. I'm pretty sure it's still not absolutely truly perfectly 100% geostrophic, or a moving parcel of air at that level would never change direction, but the component of ageostrophic wind is so small as to basically allow it to be idealized as perfectly geostrophic.
I think your struggle is that you draw $\overrightarrow{v}$ showing it changing (having it curved), whereas you draw the forces as instantaneous snapshots. Instead, try to consider each instant as the $\overrightarrow{v}$ vector turns... your $\overrightarrow{c}$ vector is also changing direction to match.
So initially $\overrightarrow{c}$ and $\overrightarrow{P}$ don't cancel at all, but as the wind turns, more and more of them cancels until it reaches a steady state when $\overrightarrow{c}$ and $\overrightarrow{P}$ are completely equal and opposite and so the velocity vector is unchanging (in magnitude and direction) as well.