Google is probably using information that it has not specifically requested of you during the password reset process in order to verify your ownership of the account. Specifically, tokens stored on your computer, and your IP address.

I had a similar experience to yours, which initially alarmed me, and tested the above theory by using the Tor browser to perform the reset. This browser redirects a Web session through Tor's own servers in Europe, making your session more anonymous.

The result was a much more aggressive set of questions. The first time I attempted to reset the password, I just blew them off, and hit a brick wall. I tried a second time, and once I had answered the questions somewhat correctly, I was presented with an emailed link to a reset page. When I clicked that link, since I have two-step verification set up, I was presented with a demand for a number provided by the Google Authenticator app on my phone. I provided that number and only then was I allowed to reset the password.

This experience gives me more confidence in the process. Google, while fallible, is not a huge corporate playpen full of idiots. Password security is a critical feature of Google's business, and I'm sure that they have thought long and hard about how best to allow legitimate users who are schleppy enough to lose passwords to get them back without allowing thieves to run off with all the Google accounts.

Direct answer

Tell your students to add your folder shared with them to "My Drive" in their own accounts and that after move or add the corresponding files to that folder.

Explanation

Files owned by the user can't be moved to "Shared with me". At the "root level" of "that place" only could be items that are owned by others could be listed.

Users could organize their own files and folders and files and folders shared with them in "My Drive".

References

Organize your files in Google Drive - Google Drive Help