Final look:
Significant loses on the diode bridge, so voltage drop after stabiliser is huge. Correct voltage on the DMM (without drop on wires). 2.5A load — no drop:
3.3A — around 200mV drop:
4.6A — 600mV drop — too much:
Around ~6.45v on the transformer (226v in socket right now), after diode bridge is around 5 * 1.4 (with capacitor) = 7v — 1.3v (regulator dropout) = ~5.7 without load (no dropout at the transformer). So with a load it’s not a 5v at all with a good current 
Bigger capacitor (now 10k uF) or schottky bridge can help. But in general 4.8v with 3.3a current is acceptable for my cases with this PSU.