I cannot imagine what discrepancy you are talking about. The EOM yields $$ \partial_ r \left (\frac{p_r^2}{2m} + \frac{p_\theta^2}{2mr^2}\right )=-\dot p_r, $$ which amounts to $$ \frac{p_\theta^2}{mr^3} = \dot p_r. $$ It is quite misleading to think of it as "the" radial force.

The radial acceleration you found first, written in terms of canonical momenta, is $$ ma_r= \dot p_r -\frac{p_\theta^2}{mr^3}, $$ so, given the EOM above, it vanishes. Where is the discrepancy?

I cannot imagine what discrepancy you are talking about. The EOM yields $$ \partial_ r \left (\frac{p_r^2}{2m} + \frac{p_\theta^2}{2mr^2}\right )=-\dot p_r, $$ which amounts to $$ \frac{p_\theta^2}{mr^3} = \dot p_r. $$ It is quite misleading to think of it as "the" radial force, as it willfully skips the centripetal acceleration.

The radial acceleration you found first, written in terms of canonical momenta, is $$ ma_r= \dot p_r -\frac{p_\theta^2}{mr^3}, $$ so, given the EOM above, it vanishes. Where is the discrepancy?