Why spin should influence the geometry of the space-time?

If you consider the connection in general relativity simply as a general connection on the principal bundle $OM$, the bundle of orthonormal frames, then it is neither metric nor symmetric (such a general condition is called the metric-affine gravitation theory).

I don't know the experimental status of the non-metric part of the connection (it's related to dilation and shear currents of the matter fields), but if we only account for the torsion part of the connection, then the Euler-Lagrange equations for gravity give us

$${R^a}_{\mu} - \frac12 R {e^a}_\mu + \Lambda {e^a}_\mu = \kappa \frac{\delta \mathscr L_M}{\delta {e^\mu}_a}$$ $$ {S^\mu}_{ab} - S_a {e^\mu}_b + S_b {e^\mu}_a = 2 \frac{\delta \mathscr L_M}{\delta {\omega^{ab}}_\mu}$$

The second equation relating the torsion tensor to the spin density tensor. By defining connections on spacetimes in the most general way possible (but still enforcing its metricity), this is the field equation we obtain from the Einstein-Hilbert action, the Sciama-Kibble equations.

Einstein's theory of relativity is defined to be torsion free not because it's the most natural condition to impose, but because it simplifies problems a lot, and, for all classical fields, it is also true. For scalar fields, EM fields and general fluids in general relativity, we have

$$\frac{\delta \mathscr L_M}{\delta {\omega^{ab}}_\mu} = 0$$

This is due to the fact that the connection on scalar fields is trivially just $\partial_\mu \phi$, that EM fields use differential forms $F = \mathrm{d}A$ which are independant of the connection, and fluids are just defined with respect to the metric tensor. In those circumstances, the Einstein-Cartan and Einstein theory of gravitation both predict the same events.

It is only for spinor fields that the spin density tensor is non-zero,

$$\frac{\delta \mathscr L_M}{\delta {\omega^{ab}}_\mu} = \frac i2 \bar \psi \gamma^{[\mu} \gamma^\nu \gamma^{\rho]} \psi e_{a\nu} e_{b\rho}$$

So in other words, the Einstein-Cartan theory is simpler in its assumptions, and the Einstein theory is simpler for calculations. For all current experimental tests, both are equivalent, but will predict different results for fermions.