Pressure fed versus pump fed rockets

The advantage of a pressure fed rocket, as I understand it, is much
cheaper (and somewhat smaller and lighter) engines, but at the cost of
very heavy tanks that hold propellant at 250-300 psi, compared with the
20-30 psi tanks used in pump fed rockets.

Would it not be possible to build very cheap low pressure pumps with 250
psi instead of the 1400-3600 psi used in pump fed rockets, so we would get
the low cost advantages of pressure fed rockets without the disadvantage of
very heavy tanks? Or am I missing something?

(Apologies if I have posted this twice -- email is acting up)

-- Larry