I gathered some statistics using the Cremona tables. There are 3337 elliptic curves with conductor $\leq 10^4$, rank 1, trivial torsion group and positive discriminant (which means that the group of real points has two connected components). Among them 446 have the generator of $E(\mathbf{Q})$ located on the identity component. This is not really enough to draw any conclusion, but here is the histogram representing the location of the generator $P$ on the identity component, in other words the real number $z_P \in (0,1/2)$ such that $E(\mathbf{Q})$ is generated by $z_P \Omega_E^+$, where $\Omega_E^+$ is the real period.

I gathered some statistics using the Cremona tables. 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.