Problems with learning curve and player elimination in Eclipse
Full disclosure: I have never played eclipse.

From what you describe about what you like and what you don't like I have some suggestions to make this game shorter and easier to learn.

House rules to the rescue!

I would start by making the game semi random. In really though draw based games where I am struggling with a multifaceted mechanic, I remove some of the facets so that I can focus on learning how to play with the others. In this case I would say eliminate an amount of 'bad' tiles/cards to perhaps focus on one facet of the game's strategy.

A game that does this flawlessly(where I got the idea) is Fluxx. With each 'flavor' of Fluxx there are new keepers and other cards added that are not core to the game. By removing them you can get the hang of the game and then incorporate the new content, modifying your strategy as you go.

This strategy often comes with a minimum of two benefits: it makes it easier to play at the outset and it also will usually unbalance the game and thus shorten time to a win condition.

If you don't like the idea of playing with fewer tiles or removing some of the bad ones, set up a limited number of times that you can 're-draw' if you truly think the card you drew is going to ruin you.(Think bennies from Savage Worlds) 

This allows you to keep most of the balance and length of the game, but allow for exceptions with new players who are struggling to learn the strategy that may be totally destroyed by one bad draw because they had no idea that horrible thing could happen and thus never planned for it..

I would also enforce a team play situation: You can not win if player X has lost the game. In another answer I made this same suggestion. Play with 'conditional' alliances until there are only two players left, then it is all out war. This has the benefit of freeing up at least two players to play something else while the game goes on.

By eliminating some of the randomness and by enforcing 'conditional' alliances, you will see that the games will become shorter and the mechanics can be focused on. Eventually when everyone has a handle on the most basic strategy, you can include all the games original content little by little.

This also adds the benefit of making the game a bit 'new' each time you play and can make it a bit more interesting to learn.