1) I believe the key is the accompanying admonition that those who are NOT destitute not "grind the faces of the poor" and the idea that those who sacrifice like the widow in that story need to be supported and helped by those who have excess. In other words, the idea is that tithing is a command for all, but those who exhibit the most faith are those for whom that tithing is a real, practical sacrifice - and that such faith and sacrifice need to be honored and "validated" by the generosity of those for whom it is not such a real, practical sacrifice. In our current "system", that means that those who can should pay a generous fast offering so that those whose tithing leaves them needing assistance can receive it without shame or guilt of any kind.
I believe in the LDS Church's institutional stance in this regard. I absolutely love that concept and principle, especially since it values the widow's mite exactly as the millionaire's millions and allows the truly destitute to feel like active contributors despite their destitution (which I believe is extremely important) - and I really wish as a people we acted that way to a greater extent than we do. If we really lived that ideal (or even came much closer to it), our world would be a much different place - a true Zion.
I understand that the way the concept is implemented too often isn't ideal - but I believe it's important for those who wouldn't be asked to contribute "naturally" be asked to contribute anyway, as long as they then can receive assistance in their need.
2) The financial situation analysis that is supposed be used in the LDS Church is something with which I don't struggle at all - again, as long as it is done properly as an attempt to help people become more self-reliant. Seriously, I have known situations where people were receiving church assistance without having to budget carefully - situations where the assistance wouldn't have been needed if the person or family would have made some simple budgetary changes. The financial analysis allowed them to do so - which kept them from taking assistance that could have been given to someone who really needed it. As is the case with MANY things, there is a huge difference between the ideal and the practice in too many situations - and this is one of those cases.
3) Finally, I believe self-reliance is a secondary goal in this discussion - in that institutional self-reliance is critical to being able to alleviate poverty for those who can't be self-reliant at the individual level. "Zion" is the ultimate goal - or, "communal self-reliance" is more important than "individual self-reliance". The poor always will be with us (meaning there always will be those who are not financially self-reliant), so those who are able to contribute to their care should do so - ideally to such an extent and degree that "poverty" vanishes even though "the poor" still exist.
How does that relate to tithing?
To me, in theory, tithing (or any other system that does the same thing, specific amount notwithstanding) is the great equalizer - in that it allows ALL to be active contributors in the community toward the building of the community and in that it provides others an objective motivation (even if they lack it instinctively) to help those who are helping the community.
There are two polarizing positions in this issue (applied to governmental discussions, as well):
1) The Lord helps those who help themselves.
2) The Lord commands us to serve and give to others, regardless of what they do with our assistance - even if it doesn't "help" them.
Both of those positions can be and are justified through our canonized scriptures, but, again in theory, I believe the combination of tithing and fast offerings is a really good way to combine them and manage the paradox in a practical way. I like that we provide assistance also to those who do not pay tithing, but I absolutely would encourage those who need assistance to "tithe" what they have and receive assistance to make up what they gave.
King Benjamin's sermon is astounding to me in multiple ways, but the acknowledgment that "the poor" can be just as proud in their poverty as "the rich" can be in their excess is deeply, deeply profound. Donating in destitute situations and then accepting help from others is a great way to mitigate and eliminate that natural pride of the poor - and I can say that, having been in that situation more times than I wish.