There is a third option besides God having a purpose for sickness and God not being able to eliminate sickness - and that is the church.Do you see where your beliefs bring you? By saying that God has no good purpose in sickness (though you allow that He may have good purposes in other—even more torturous—sufferings, so long as they come from evil people, not germs), you have created an artificial limitation upon God. He would like to destroy all sickness (since He can find no good purpose in its existence), but He is helpless to do so!
If we were to say, “God would like to end persecution, but cannot do so prior to judging the whole world,” we might have a scriptural case, since persecution springs from man’s free will, which God might have a policy against thwarting prematurely. But sickness, in most cases, is not produced by human free will or agency, and would not seem to have any innate claim on special amnesty against being annihilated by God at His pleasure.
This may open up a separate theological discussion - on whether or not we Christians today can view the anointing and calling of the apostles as being for us as well - but Jesus commissioned the apostles to heal the sick as they preached the gospel:
"Jesus summoned His twelve disciples and gave them authority over unclean spirits, to cast them out, and to heal every kind of disease and every kind of sickness. "(Matthew 10:1)
What about those people with sickness that God wanted to be sick? Surely they didn't have authority to heal those people. "Every kind" seems pretty clear that "every kind" of sickness was a candidate for them to heal.
Anyway, I'm not sure if you believe Christians have this same authority today - but I don't think we have any less of a call then they did. Jesus wanted to "multiply himself" in a way, and reach more people with the gospel than he could by himself - and at the same time, train them for when he would no longer be on earth.
That's the role we have today. We are figuratively and literally his body. The Head has desires for this world and his body carries out those desires - however, if the body fails to act, there are things the Head wants done that won't get done. God is sovereign, but He has sovereignly linked his plans with man.
Whether you agree with this notion or not, it is at least a logical third option - that it is God's will to heal everyone who is sick - but that he has employed the church to carry out this decree.
In a similar vein, not all suffering is good or God's will. God's desire is for us to seek justice, defend the cause of the orphan, and help the widows. (Isaiah 1:17) There are forms of suffering that God does not want to occur - but he has made us stewards of this place. If these things are not corrected, it is not because he is powerless to help, nor that he has a greater purpose for orphan suffering, but because his appointed and anointed are not being obedient to carry out his instructions.