The impossibility may be "unacceptable" but it's reality. I believe that at a certain point - e.g. when evicting one means 1000 or 10000 more people are able to afford somewhere decent to live near where they work - eviction becomes a lesser evil. If you're unwilling to make that kind of trade, then however much sophistry you put into "focusing on solutions that solve both sides", more likely you'll just end up never getting those 1000 or 10000 people a place to live.