Capacitive filtering draws short bursts of current. To get 5A output you must supply bursts of current on the order of 30A to 60 A. I*I*R losses are high. Heating effect is high, through diodes and capacitors. Your transformer may or may not limit these extremes to a reasonable level (depending on internal impedance, saturation ratings, etc.).
However if you add a choke filter it lengthens those bursts and reduces their amplitude. A mere 4mH is sufficient for a draw of several Amperes as your supply must handle.
A full diode bridge is not essential. Two diodes are sufficient. Ground is one end of the secondary. The split volt levels stay better balanced even if loads are unequal.
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