I like the idea of using a page that keeps getting overwritten. One crude workaround for the message variants issue would be to just wrap the raw message as entered by the sender in <pre><nowiki>, since the output really isn't all that important for purposes of record keeping.

Messages can also contain substed templates which is difficult to represent accurately in a page revision. (Of course, a log of message texts would be useful even if that bit would be inaccurate.)

I would support keeping track of messages this way if possible. For the record, the log entry referred to is defined in the function pasted below

public static function logToWiki( Title $spamlist, User $user, $subject ) {
		$logEntry = new ManualLogEntry( 'massmessage', 'send' );
		$logEntry->setPerformer( $user );
		$logEntry->setTarget( $spamlist );
		$logEntry->setComment( $subject );
		$logEntry->setParameters( [
			'4::revid' => $spamlist->getLatestRevID(),
		] );
		$logid = $logEntry->insert();
		$logEntry->publish( $logid );
	}

One idea is to have a page such as "Project:MassMessage archive" that gets overwritten with each delivery, providing a record of the message that was sent using a permalink. Or perhaps use "Project:MassMessage archive/dated/subpage/id" as a page title format instead and don't overwrite.

Another idea is to unconditionally post the message being delivered to the initiator's talk page, which would also provide a usable permalink.

I also like either idea. As Tgr notes, the wikitext can be anything, so the text would need to be in <pre> tags , or otherwise unparsed, so that the entirety of the submitted wikitext is shown. Both the sender's user talk and an archive page may not be suitable targets for the specific wikitext, for instance if it relies on templates which don't exist on the sending wiki.