"Are [the rivets] just holding the spreaders in place until the shrouds are installed?" someone asked.
(Not just random "someone", but someone with a 16' catamaran with a simpler rig. A boat that can easily sail circles around Red Ranger. But doesn't have a bar or heads.)

The hardware is a stainless "base" or "sleeve" and the aluminum spreader arm.
There are a handful of reasons to rivet the base to the mast. I enumerate them here because it's really important for me to think these things through in some detail.
(The whole going to sea and being self-reliant demands a certain amount of knowledge about Red Ranger and her various pieces and parts.)
Here's my understanding:
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As asked, it holds the spread in place while replacing a shround. Good call. Yes. (The mizzen was taken off to fix the deck, back in '09 or '10. The shrouds were slacked when we replaced the chainplates.)
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When running downwind, one set of shrouds (and backstays) are lazy. Since they're doing nothing to hold up the rig, you want them to stay in place anyway. For example, to ease the boom out to starboard, the starboard backstay has to be released to get it out of the way. The mast is actually pulling against the port shroud and backstay as well as the sheet. The starboard spreader is slack.
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It prevents the hardware from "walking" under cyclic loading. When Red Ranger is rocking gently at anchor, and I'm looking for the bottle of good rum in the bar, while trying to keep from spilling my cold tonic water, I don't want the base for the mizzen spreader to be shifting its position. I have important rum to drink while at anchor. Or perhaps I finished the rum yesterday, and we're back to beer?
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I can stand on it when ascending the mizzen.
So the rivets are important, but a little goofiness in one is no reason for alarm.
I have replacements. I have the Marson Big Daddy. I've drilled out screws and rivets before. I (finally) bought the fancy titanium drill bits for drilling metal. I can get some Marine 31 or ZEP polish and clean it up.