Not a direct answer, but:
Do they sound like they are failing? This may seem silly, but older switches might have fans that sound really bad, and that is a strong argument right there "Come listen to this."
Compromise, a Cold Spare:
Also, another way to approach this would be to argue that although the downtime is not worth it to them implement full redundancy, but what about the price of a single switch that you can have on the rack as a cold spare. This way if one does fail, your recovery time will be much less time.
Joel Spolsky mentioned this in one of the stackoverflow podcasts, something like "Recovery time is more important that how often it goes down." The argument being as I remember it, that downtime isn't as big of a deal if you are back up in a couple minutes, but is if you are not back up for half a day. Smart way to look at it in my opinion.
So a your new argument might be that since switches are not that expensive, it is cost effective for the business to have at least one cold spare because it can make the different between being down for minutes instead of a whole day.
Also if you win your argument this way, be sure to pre-configure the cold spare ;-)