Mozilla was the original code-split from Navigator, and it's purpose was to preserve Navigator as a browser for the half of the web that was optimized for it (remember the old "best viewed with..." buttons? Good days). Firefox née Phoenix was a fork from Mozilla to strip out Netscape-sponsored features of the Mozilla engine (giving us the Gecko engine). It succeeded in this goal, as well, for a time.
Your history is a bit off. Gecko was Mozilla's focus since Mozilla itself was created to continue Netscape's work on the next version of their browser after failing on their goal of improving the (horrible) Netscape 4.x layout engine, which was their original goal for version 5 (although I think they might have been experimenting with both possibilities at the same time before giving up the former). Firefox (originally Phoenix then Firebird) was created with the goal of taking that same layout engine, Gecko, but wrapping only a simple browser around it rather than the entire Mozilla/Netscape Communicator-style suite. Netscape never had many Netscape/AOL features in the Mozilla suite itself; those (e.g., AIM integration, branding, and a different default theme--Modern instead of Classic, etc.) were mostly confined to the Netscape-branded releases that AOL/Netscape released using the Mozilla suite as a base (starting with Netscape 6--skipping the scrapped version 5 attempt, though version 6 was horribly delayed and based on a somewhat unstable pre-1.0 release of the Mozilla suite). In any case, Gecko has not only been there since before Firefox, but it's one of few things that Firefox and the Mozilla Suite (which effectively lives on as Seamonkey) share, albeit a very large and important thing since it's used for so much (not just HTML rendering but also creating the UI itself via XUL and a theme).
Thunderbird was created with a pretty similar goal: take the same layout engine but include only the e-mail features from the suite.