What you are describing is too dramatic
NP100s are extremely reliable and they are not that old
the turn-on timer is triggered by a 555 timer that derives power from the 6.3v filament supply.
pull out 1 of the 6sn7 tubes and see if it turns on after 60 seconds and stay on
if so you may have either a tube that has a filament shorted therefore drawing too much current. Or a filament supply cap 16v 10000uf that has deteriorated, or one of the white 3ohm drop resisters next to the filament supply caps that have bad solder joints.
there is no thermal protection on the np100, only a thermal bias reducing circuit that doesn’t cut off as dramatically as pulling standby or not popping to “green” operating mode.
there is enough urban folklore on counterpoint SA amps and NP amps it’s disgusting. No putting an ice pack will only get the amp wet and explode. Do not attempt!