Do we connect the ground rails on both sides of a SMPS?
Regulatory agencies require galvanic isolation between the primary-side and any user-accessible voltages; this is provided by the transformer, maintaining creepages/clearances between primary and secondaries, etc.

Capacitor connection between primary return (the rectified low-side) and output return is done for noise attentuation, as you have observed. This is often done with two agency-recognized capacitors in series, so that if there is a single failure you still have a recognized component providing some measure of safety.

Often the low-side of the user-accessible output is tied to earth/chassis, again for noise reasons (and to have a solid reference point). Tying the bulk DC low-side to the output in this circumstance also ties it to earth, which (if your AC distribution is using a neutral) means that you're shorting your bridge out. This is a bad thing.

(Inexperienced or sleepy people sometimes inadvertently simulate this scenario by touching an earthed oscilloscope probe return to the bulk DC low side. Often the scope protects the power supply fuse by blowing itself to smithereens first, or turning the scope lead into a toaster element by glowing red, but YMMV)