I agree, they can be enjoyable. In fact, I love that when you get engaged in Alpha Trianguli against the Terran Empire, you can't get out until you beat them.
However, I certainly agree the auto-engage creates alot of issues. My girlfriend got auto-engaged in the Orion Sector (generally a Lieutentant level area) but when she got pulled in, she was instantly vaporized by a +37 Klingon Neg'var Battleship.
I would like to see DSEs changed so that the award reflects YOUR level, regardless of where you engage, and I would like to see it be something you seek out, and not something that jumps you while you are trying to travel.
No they canot be removed because players need them to lv up because the missions alone cannot get you high enough to get to next lv.
Unless you're crazy about play super efficient (which, from what I can read, doing DSEs only appears to be the fastest way, even trumping quests, to level), I find doing Explore missions to be better: You get XP and Badges to get better gear.
Actually, I'd like to see auto-navigate build in: So for example I tell my ship I want to go to DS9 and it keeps switching maps merrily while I go get a coffee.
In this scenario, auto-engage is totally worth it - do you risk the coffee, or do you stay at the helm to execute evasive manoeuvres
On a real ship, you could tell your helmsman to avoid all enemy ships on the way to DS9 while you get the coffee. Although, actually, on a real ship, you'd get it delivered to the bridge. :-)
But still, you could have standing orders with the helmsman to auto-engage or not, which leads me to think the best situation would be to have it be an option (I would think default would be auto-engage off) in the option menu to turn auto engage on or off. It would seem to be an easy thing to program.