Based on #19851. These are just ideas and the exact endpoint values are open to discussion.

GET /session/:sessionId => GET /session/:sessionId/appium/info
GET /session/:sessionId/log => GET /session/:sessionId/appium/log (already exists in Selenium as /se/log)
GET /session/:sessionId/log/types => GET /session/:sessionId/appium/appium/types (already exists in Selenium as /se/log/types)