This doesn't sound right for a few reasons.
First Godaddy manages a massive amount of domains. For them to be doing anything like this would really defy common sense. They'd be hit with tons of complaints and importantly so would ICANN. AND ICANN (from our experience) will send out a notice (which creates paperwork for us) on any complaint literally for any reason. You have to respond to it so they clear the complaint. It's a pain and separate from any other reason a big incentive to not have people complain to ICANN (which can happen even if you are not at fault). ICANN also audits from time to time and they will pick a number of domains and make you show logs that you have sent out notices. (But my initial point is the more important issue).
Further ICANN requires that registrars send out multiple email notifications. They specify a number (I think it's about 4 or 5). So it doesn't make sense (especially for Godaddy) that this could have happened '1 notice 2 hours to respond'. And only a single notice.
As far as the cost, registrars are charged by the underlying registry (.com/.net .org .info) to bring a domain out of redemption. That charge is about 4x roughly what a .com domain costs for a year so it's not trivial. Then that charge is marked up by most if not all registrars. Our fee is less than what is stated here but it's not nominal either.
Separate I have a few of my own domains at Godaddy (and some other registrars) 'just because'. I always get plenty of notices from them it's almost annoying.
One last point. ICANN requires a domain to be put on hold prior to going into redemption. There is no doubt that Godaddy is doing this. As such your first notice of domain not working would be it's on hold not it's in redemption (and set to delete). Seems that this domain was not being used by the OP and they didn't pay (which they acknowledged) and it was deleted and they didn't notice it was on hold (specifically to get domain owners attention). I don't know what the 'off hold' fee is that godaddy charges my guess is it's less than the redemption fee.