I had to replace a spindle on my 4500 and also saw that video. I had to take the deck off; really not that hard. The pulley nut on mine was a night mare. Good luck.
FYI there are some old posts under spindle replacement for the same model and they all had the same trouble as I experienced.
If you need a manual I posted a question about a belt issue I am having with a similar title and one person sent the link for a manual you may find useful; I did not have one.

I feared that the nut on the pulley might be the problem since it is not accessible with a heavy duty impact with the deck on the machine. Does that sum up the reason you removed the deck?
I do have the Owner's Manual and downloaded to My Documents one that I found online. But are you referring to a repair manual? One certainly must exist for the service centers. And AYP might include it with mowers sold under their brand name. I do not know the AYP model number for the same mower.
Which manual did you get? Do you have a link for it?

Shopping for price, I ordered a new spindle assembly from Amazon, to be shipped by MowtownUSA. Bought on May 31, they note that they probably will ship by June 7. They tell us those dates after we complete the transaction. We buy these parts because we need them this mowing season, not next year. .
Searching for online reviews shows that Mowtown is not good at all, not good BBB rating. So I bought one on eBay June 3 and expect to be mowing before June 7.

I used the blow ups on the Sears parts site. FYI since time is money in my book; they get you the parts in 3 days.

For your Sears parts, just Google-
AYP part#