Also, if this game was re-made, what parts of it would you definitely want to see again? What parts would you be more than happy to get rid of?
I'll start. When I first started playing online (going beyond LAN games with my siblings), the words felt real and solid. I could jump up onto blocks, fall off of them, drive into them (and they would stop me), and ricochet shots off of them. What you see was what you got. There were no complicated physics that would be difficult to figure out. Also, the worlds I liked (overlord and babel, mostly) were big enough that you could run around and explore or feel it out a little bit without having to be constantly in the action (unless someone was hunting you). Even though I had to learn new skills for years, I felt from the very beginning like I understood how it worked (at least once I figured out how to lock on with a Guided Missile

What would I change if we started over?
While all the BZDB settings make for some variety, some people have used that to completely alter physics and core gameplay elements, which means some skills (like precise movement, dodging and timing of shots) don't transition over between maps. I would prefer that very few gameplay dynamics be modifiable (say, for example, the wings jump count, and perhaps allowing the Shield flag to take a certain number of hits, and maybe a few other things like that).
I would also probably want to eliminate most of the 2.0+ map objects. There have been a lot of maps that have better eye candy than actual playable designs. The one useful feature of meshes I can think of is one-way shoot-through panes (like on Missile War), but the same could be done with a teleporter or zero-thickness box. Also, with three different kinds of solid geometry (boxes, meshboxes, and meshes), each with its own physics behavior, there is a lot of inconsistency in behavior and some problems that have come up (like trying to teleport from on top of a box to on top of a mesh). During BZFlag's heyday, it was generally the 1.x maps that were the most popular and seemed like they worked the best. If the game only had a single set of basic geometry types (the 1.x objects, with some minor enhancements), then we could focus on making those objects more graphically detailed and visually appealing, and avoid so much complication. Oh, and I would fix the glitch that lets you shoot through the ramp or through the base on HiX.

The last few things would just be some basic modernization and refinement. This would include standard input controls and settings that actually work for competition, a simplified menu interface that accepts mouse (and maybe even touch) input, improved atmosphere/weather visual/audio support, and implementing LUA script support for things like AI bots, tutorial mode, server/client customization, etc. This would also include some internal enhancements like making world initialization more straightforward and OOP-based (so you could launch a game from the client without needing the external server binary), some kind of better solution for lag than we have now, and some other icing on the cake. Oh, and of course a complete re-do of the world/tank/effects rendering using the more modern shader pipeline, which should reduce the performance cost of more detailed graphics, make the game more visually appealing for modern tastes and make the game more future-proof.
What would your desires be?