Thanks @Yarnabrina, I phrased that somewhat poorly. What I meant is that it's very unlikely to be a problem with R Markdown and much more likely to be an issue with syncing via Google Drive.
The R Markdown process won't delete files 5-90 seconds after they're created; once it stops running it will not (can not) change the files. If RMD is the problem, the file will be deleted during or at the end of the rendering process. If there's a delay, then it's not R Markdown.
The issue with files changing spookily is consistent with file-syncing services. Maybe the Windows and Ubuntu machines are fighting with each other? Idk but there certainly are a lot of hits on Google for "Google Drive files disappearing".
Edit: @Yarnabrina's suggestion to try pausing syncing is a great diagnostic test idea.