A question about heating of a slowly turning off MOSFET
If I use the procedure I wrote at the beginning the temperature rise becomes
ΔT = P × θ = 30 × 62 = 1861°C
But this is insane. ... Do I need a heat-sink here?

You are neglecting to consider the heat capacity of the chip. This acts like a capacitor in the thermal equivalent circuit, connected between Tj and (some arbitrarily defined) ground, which prevents the chip from heating up instantaneously. Unfortunately you are unlikely to find good data on exactly what the value of this capacitor should be.
You can refer to the safe operating area curve for your MOSFET:

Based on this it's probably safest to keep your switching time below 10 ms, instead of close to 1 s as you have currently.