A secondary objective of the Ds in the shutdown -and acknowledged by them but not promoted due to wonkiness - is to rein in the impoundments and rescissions - ie Trump not spending budgeted money - which has been a huge underreported issue for the country.
The bigger thing, though, is that I'm not sure what Democrats' strategic objectives really are, beyond saving Schumer's job/boosting base morale. Tactially, it seems to be going pretty good! But tactics are not strategy.