I gathered some statistics using the [Cremona tables][1]. There are 565803 elliptic curves with rank 1, trivial torsion group, positive discriminant (which means that the group of real points has two connected components) and conductor $\leq 4 \times 10^5$ (the current limit of the tables). Among them 239106 curves have the generator of $E(\mathbf{Q})$ located on the identity component.

Here is the histogram representing the location of the generator $P$ on the identity component for these curves. More precisely, this is the distribution of the (unique) real number $z_P \in (0,1/2)$ such that the Mordell-Weil group $E(\mathbf{Q})$ is generated by the class of $z_P \Omega_E^+$, where $\Omega_E^+$ is the real period of $E$. We can note the behaviour near $z=0,1/4,1/3,1/2$ which might be explained by the fact that the torsion points somehow repel the point of infinite order.

[![enter image description here][2]][2]


  [1]: https://johncremona.github.io/ecdata/
  [2]: https://i.sstatic.net/eWqsM.png