simple DCDC Step Up converter mystery
I noticed that when trying to power the amp board, SW out of the
controller to the mosfet is 3.5v that seems a bit low to make the
mosfet fully conductive. Though this signal can't be more than Vd and
if Vd=Vout=Vin how is it supposed to start the system?

IRF630 has RDSon of 0.3 Ω with 10 V Gate drive, and needs ~6 V to turn on fully. 3.5 V is indeed too low to turn the FET on properly. But even if it did turn on fully its Drain-Source resistance would be a problem because at 2.5 A output the input current must be &gt; 2.5 * 6 / 3.7 = 4 A, but the FET would drop &gt; 4 * 0.3 = 1.2V so it would need to draw even more current to get the required output (which would drop more voltage, which...).
Bottom line is you need a FET with lower Gate drive voltage and lower RDSon.