I would much rather get a mode where you get:
1. A set of pre-set rifts that you can pick from.
2. A pool of pre-set items, let's say 2-3 for each slot that have some synergies between them, but not enough to create any of the popular builds.
3. You can't reroll stats, find extra items, etc., so you are stuck with the items you are given, but you can freely set up your skills.
4. While the each of the pre-set rifts has static layout and monster distribution, you are free to increase the difficulty level.
5. The idea is to make the most of what you're given (both the items and rifts) and see how far you can go with it. You win (i.e. get the cache) if you manage to beat a set difficulty level on any of the pre-set rifts with your build. Once you do that, you get an extra, smaller cache for each couple of levels above it that you can clear.
6. You play with your character - the only difference is that you don't have access to your items and stash, so that you have to build with what you're given. However, you normally get experience and it's carried between CRs and normal game.
With the base reward level set at a moderate level, everyone could get their weekly cache by either creating basic, fairly logical build or even following meta builds (I'm sure people would post them online), buf if you're one of the people who likes challenge, you could compete with others by clearing as high level as you can as fast as you can - dependent not only on how well you can execute a build, but also on how well you can create your build and adjust it to one of the rifts you're given.
This removes all the problems you've listed OP:
1. You create your own build, so you can create something that you find fun instead of playing with something someone else has created that you might not consider to be fun. With the base completion difficulty set at a moderate level, you wouldn't really be required to min-max to get the weekly cache, so there would be a selection of builds that allow you to complete the challenge - thus, you could pick/make one that you feel is fun.
2. The rewards scale with progression - though you do not get much for going higher, if you like this mode you can absolutely try pushing, if not you can just take the base cache and go play your normal character.
3. Only 1 class per week - sure, it would still be only 1 class, but with the much wider choice of builds it would be much easier to play something that you find fun and that fits your playstyle.
4. It involves pushining - but makes it completely optional.
5. As you can adjust your build, you can make it less susceptible to being f...ed by RNG.
1. It's much more varied than the current system as you get an option to play a wide variety of builds, not just one.
2. There's much more gameplay in it - you get a couple of rifts, a chance to mess with the build, an option to progress instead of just playing 1 pre-set build on 1 pre-set rift. I feel that this alone would make the mode much more interesting and lead to a weekly meta being built around it.
3. It's less frustrating and more personal - you build your character and play it, so if anything goes wrong you don't feel like you're screwed by some "random noob who just put a bunch of random items and skills together".
4. You might not progress your character as efficiently as while playing the standard game, but you still get at least some mats and experience.