I have a solenoid type espresso machine which seems to be completely
blocked up

I can hear the pump pumping and water comes out of the wand so the
pump is clearly working but nothing comes out of the head. I have
checked the solenoid and the last chance for this expensive machine is
cleaning out coffee or limescale in the system. I took the head screen
above the basket out and it is completely blocked as are the various
holes in the piece above the screen. When I stuck some pins in the
channels I did manage to get some drips out of the head. But there is
no real relationship between the pumping and the drips.

What I was wondering was which cleaners will burn through all this
crud. To me the citrus ones all seem to be the same, none seem to
be of any greater strength than others and I am not sure whether they
are designed to dissolve the coffee or clean out limescale in the
boiler. Whenever i have tried them in the past I have always found
undissolved crud under the screen after use and what has come through
has never seemed particularly dirty to me??

Lastly is there a balance to be drawn between trying to push cleaner
through clogged pipes and pumping so much that there is a risk of
blowing the pipes/pump/electrics?