Sorry, I misled you. That "Download" option actually refers to downloading to their server when Magic Backup is turned on. Label importing happens automatically when you're restoring from their servers using
Magic Recovery.Thus, their label exporting/importing functionality is only useful for users of their Magic Recovery service, and not beneficial to others. I'm becoming disillusioned with them.

Perhaps they will make further changes to this functionality in the future. This is the first Envoy version that supports label exports from hot wallets at least.
I thought about tagging @foundationdvcs and sending them a PM to check our conversation and shed some light on all this but I can see that they haven't been online since May. I will get in touch with their customer support again and ask. In the meantime, I will make an update in my OP to mention that label exports only work with Envoy (hot) standalone wallets.