The application will not prevent you from selecting a Triggered Send-Templated data extension as your entry source as it is simply a sendable data extension. I don't see any system-related issues that would occur using a setup as you have described.
However, depending on the use case it could work, or querying into another data extension may be better suited. For instance, if you want to send a series of emails to those that were sent a triggered send, then this setup could work. However, a potential downside to this setup is what if a triggered send errored and caused queuing? You now expected those subscribers that were supposed to be sent a triggered send but didn't get it, then the following emails may not make sense to the subscriber.