Is it possible to reset the sequence of indices that vim uses for file buffers?

No, not without restarting Vim. There are a number of reasons why they cannot be reset, some of them internal, some having to do with avoiding surprises with Vim scripts that store buffer numbers.

Are you aware that you can do :buffer some_partial_buffer_name to jump to another buffer, as long as the string you supply (some_partial_buffer_name) is unique?

Creating a session (:mksession!), closing vim and opening the session (:so Session.vim) will reorder you indices starting from 2 (1 is used to source the session).