On 3 different Apple M1 / M2 computers all running Ventura 13.5.2 and ALL machines using the same iCloud Account as the Only e-mail server.
Add in an iPhone running iOS 16.6.1 - too
Using the same Configuration as yours - can not explain why your partners is not but the Send Messages do appear on all the devices
Then have a few suggestions ;
1 - Get Support Choose a product and we’ll find you the best solution.Start now and open an Apple Support Ticket as they are Apple Employees to deal will these types of issues .
2 - Product Feedback - Apple and make it known to Apple regarding this ongoing issue