Here’s my dollar two ninety-eight:
I’m a slowpoke mechanic. I look, study, look at the FSM, squint, study some more, peer at it sideways, eyeball it again up and down, sip some coffee, take a break, look it all over again once more, consult MUD, and then slowly start removing fasteners. To keep track of the fasteners, I try to thread the bolts back into the holes they came out of just as soon as the removed part is clear. That helps me to keep track of things.
It took me a good long time to remove my ’62 radiator, but I know I could do the job faster now. I was advised to replace the water pump at the same time. I did not heed the advice. Two months later, I was back in there replacing the water pump!
I’m with the guys here who advise cutting the fan shroud in half horizontally. I did do that and it is good advice. I’d also recommend that you replace as many hoses as you can, plus the thermostat and the water pump while you have everything all pulled apart. I’ll go one further; flush out your heater core, too.