You are over-analysing this:-
Your cheap meter should be able to see around 340vdc,even if it has a fair bit of ripple on it.
When you are seeing 3v, or whatever on the bridge DC output,switch your DMM to ACV & check the Mains voltage applied to the rectifier.
If there is a high resistance connection to the fuse,this will show up as a reduced AC input to the rectifier.
Another way to do this is to remove the PSU from the Mains Supply,& check the resistance between the two Mains input points & the corresponding rectifier pins.---It should be a low value,& both legs should be similar.
If all is well there,then remove the 1mH inductor & its parallel 4K7 resistor---this will isolate everything to the right of these components on your schematic.
Recheck for 340vdc.------If there is nothing there,it looks like your bridge rectifier is faulty,or the 4.7uF cap is effectively short circuited.
When you are testing with your "cheap" meter,remember these are fairly serious voltages you are looking at,so check the ratings of the meter,& in any case,try to make any probe connections prior to turning the PSU on,rest the meter on the table rather than hold it in your hand,etc.