Ok, you didn't answer my question about "Do you have another device on the old 500 that uses the menu button so that you can check to see if the button is actually working?" If the menu button works for another component, then the issue isn't with the button itself but the programming connected to it for the Sunfire. If the programming was lost on those buttons for Sunfire control , you won't see the purple flashes.  If you don't then can you temporarily add a device and learn a command to those buttons to test?

If the receiver came with a rebranded mx-500 then it must be around 2010 or earlier. I use MX-850's (PC programmable) and looked at the software. There are commands for Sunfire Theater Grand I through IV, TGR-401 along with 8 other groupings of Sunfire codes. I checked a few and they do have codes for the Menu button. Now whether or not the codes changed between models of their receiver, I don't know. However for other manufacturers (such as Yamaha), they generally use the same codes unless new functions are added on newer receivers and then new codes are used for that.

To try, you would need an MX-700, MX-800 or MX-850 (last iteration of the remote) and the software. The software can be found here: [Link: files.remotecentral.com]

It was for a NX Technologies receiver remote that used a rebranded MX-800 but is the same software used for MX-700 through the MX-850. Buying MX-700's, 800's & 850's through eBay can be a hit and miss.  Lots will be well used and may no longer work (rubber membrane issues or dropped numerous times and the backlight no longer works).  Some may be listed as new.  There is one listed now as New (other) with condition as Open box.  Extremely reasonable price.  This is not my listing nor do I know the seller.
[Link: ebay.com]

and a Buy It now reasonably priced:
[Link: ebay.com]

There may be some others but i stopped looking.
They require a serial cable (be sure it is included in what you buy) and if your computer doesn't have a serial port then you'll need a Serial to USB adapter.  The one most recommended for these remotes is the Tripp Lite Keyspan USA-19HS.

The good thing about getting something like the MX-850, the software and "program" will be on your computer should you ever need it. (I'd back both up somewhere too :). Also, if you change any components the remote will still be good and can be reprogrammed.  If the codes are not listed in the software database, you can learn them to the MX-850 and save them along with the program.  So you won't be out anything if you replace the Sunfire receiver.  And as a good thing, the buttons are located in the same places - with the exception of a couple that have been moved slightly.

If the Menu & #3 buttons work for other components, then I think you may be out of luck with that remote.  A universal remote may be your only option unless you contact the manufacturer and they happen to have one of the remotes.  Also, don't try to get a URC remote other than the 3 I mentioned because they will require software (Complete Control aka CCP software) that you will not be able to get.

Feel free to PM me if needed.