How come DBD::SQLite2 does not produce a 'sqlite' binary?
Because it's not supposed to, and doing so would be redundant.
The sole purpose of DBD::SQLite[2|<suffixless>] is to be a plug-in module for the Perl DBI framework, so you can use SQLite via the DBI programming interface.
The DBD::SQLite distribution leaves out a few files that are in the usual SQLite distro, including the shell and TCL wrappers, and provides an XS/Perl wrapper instead.
If you want a sqlite binary that is used independent of Perl, then you don't use DBD::SQLite on CPAN, but rather the usual distro on sqlite.org.
-- Darren Duncan