No question that drain will help. But, later if you find your basement is still flooding, either live with it or get gutters.

Now, to correct you on a couple of things about gutters-----they have little if anything to do with ice dams----I read up on this when my wife's uncle was having trouble.

Ice dams are caused by differences in roofing over an attic (where snow could melt) vs. roofing on overhangs/eves, where there is no heat-----Ice/snow melts on the warmed section and re-freezes when it hits the cold---backing further melt water under the shingles.

Now as to cleaning gutters----if you ever reach the point where you give in----I've found the best aftermarket gutter guards that keep out leaves and are easy to install, and remove if you ever need to.