Since it appears you think it could be fluid pressure related, when cold and before muscling the steering wheel, if safely able, pop in N and see what your brake pedal does. Is it super hard, doesn't want to budge? Don't try to force it down just see if it feels normal. If it too is hard like steering then you'll know if it's whole fluid system, if not then just in steering circuit. Cold does weird things to metal parts, check & psi regulating valves causing them to stick. I'm thinking you issue might be internal to the pump, I don't exactly recall for sure, but I think the pump discharge port has a spring that holds a ball to maintain pressure discharge. Perhaps the ball has a worn in ring on it that causes the ball to stick not allowing psi out for a period of time?
You should be able to place wrench on pump port where line connects then wrench on line nut and remove line, if you can plug hose you'll be able to reduce leakage and possibly not have to go thru entire system bleed process. Once line is free the port on the pump the fitting/port/regulator valve can be removed, careful not to damage reservoir on pump when wrenching port off, a quick firm tap on wrench should break it loose. Then you should be able to visually inspect parts. Check FSM for possible trouble shooting procedure. Some years back I saw a thread here describing stock regulator spring height and stretching the spring to a suggested height would provide more steering assist. Don't think you need "more assist" just need to figure out what is sticking. I think the brake hydro unit may have a check valve too? Check FSM before doing anything, maybe you can find an online exploded view of components that'll show check and reg valves. Hope this helps somewhat