Short-circuit primary side of a transformer to collect all the spikes at the secondary

What you may be getting confused about is the "ideal transformer" in the equivalent circuit. You should not regard it as having any magnetic qualities at all. Try and see it like this: -

Whatever voltage you have on the input to the ideal transformer, \$V_P\$ is converted to \$V_S\$ on the output of this "theoretical" and perfect device. It converts power in to power out without loss or degradation such that: -

\$V_P\cdot I_P = V_S\cdot I_S\$

The ratio of \$V_P\$ to \$V_S\$ happens to be also called the turns ratio and almost quite literally it is on an unloaded transformer because there will be no volt-drop across R3 and L3 and only the tiniest of volt-drops on R1 and L1.

This means you now have a relatively easy way of constructing scenarios of load effects and recognizing the volt drops across the leakage components that are present.

The equivalent circuit of the transformer is really quite good once you accept that the ideal power converter is "untouchable" and should just be regarded as a black box. For instance you can measure both R1 and R3 and, by shorting the secondary you can get a pretty good idea what L3 and L1 are. With open circuit secondary you can measure the current into the primary and get a pretty good idea what \$L_M\$ is too.

I think your expectations are wrong. If the primary and secondary are isolated, you should expect to see some random voltage develop between them.

If the voltage between them were consistently 0 V, that would be a sign that they are electrically connected and that the isolation has failed.

As another answer points out, you may not want perfect isolation in order to protect the secondary from damage due to electrostatic discharge. But even so you might only constrain the floating voltage with back-to-back TVS diodes or a spark gap, which would still allow the voltage between the two sides to vary by 100's of V.

And many isolated systems won't have any such coupling at all. In fact if the isolation is for safety reasons they will likely have to pass a "hipot" test to show that 100's or 1000's of volts can be applied between the primary and secondary without appreciable current flowing.