While I was digging through the Flash drive to see where the "Xserve" was defined, I noticed that "shareAvahiSMBModel" variable, in the /boot/config/share.cfg file is set to Xserve. I changed it to use MacPro7,1@ECOLOR=226,226,224 instead (which matches with my config in Extra smb parameters in the webui).
Then, I went ahead and modified the currently "loaded" avahi config, /etc/avahi/services/smb.service, replacing the "_device-info._tcp" to use the same MacPro7,1 flag. This fixed all the issues with the sidebar icon in Finder. Haven't rebooted yet to verify this config is loaded on boot.
However, I had another issue where connecting to the SMB share with the iP address (using say, smb://192.16x.xxx.xxx/ShareName) created a duplicate in the sidebar (one with the broadcast Avahi/mDNS/Bonjour name, and one with the raw IP address). To fix this, I changed my smb connections to use the Avahi url directly, smb://server_name._smb._tcp.local/ShareName). Now it works well, and Finder doesn't have a generic file icon anymore.
For context this issue occurs because Finder moved to using SF Symbols in the sidebar after Big Sur. Unfortunately the Xserve / RackMac icon which Unraid uses by default has no matching UTTypeSymbolName mapping inside SF Symbols which is why it breaks just for the Finder sidebar. (Source: Reddit discussion about this)