You're right. Actually that string isn't just a drone, though it makes a good one. Since the guitar doesn't actually change tuning, normal (standard tuning) chord formations can be used which utilise the uncapoed string so that drone string can be used as it normally would and as a drone. However, the real benefit of partial capoing is the ability to use more convenient chord voicings.
For example, with the drop D capo leaving the low E string open and the other 5 strings capoed at the second fret, fingering and playing the D major open chord shape sounds an E major chord with the added benefit of the low tonic root on the sixth string. This sounds much like a regular E chord except it has the third as the highest note of the chord, and the fifth within easy reach. This is not enough to justify the technique to some folks until it's pointed out that with the same capo positioning, the fingering of a G chord, which will sound an A chord, can be embellished with the open sixth string which is the fifth interval of the chord.
Also, melody notes played with a low root drone or harmony are one thing, but consider that in the key of E, as in our example with the partial capo, all the open notes are in the key and therefore more easily used for harmonizing. By contrast, to play in the key of E without a capo as described means that the open third and fourth strings (G&D) are not in the key and need to be fretted or muted or avoided. Capoing like this prevents the accidental playing of open...um...accidentals. This is a common feature of such capo use.
Without a similar description of each altered capo, I think there are several other common features. Not only can a drone be used, or a more convenient root note, but chords too, like the five (power) chord created by playing the fifth and sixth strings of our example. The sixth string fretted at the seventh fret is the same pitch as the capoed fifth string which makes for a stronger bass on an A (sounds a B) chord.
I used this example because the key of E open chords don't lend themselves particularly well to open strings and "add" chords while key of D chord shapes do. My humble opinion is that this way you can have the best of both worlds, or keys. Gets away from playing a B chord too, which lots of folks don't like.
Not to prolong this, but each different capo has its strengths and weaknesses, as do various tunings, and partial capos can be used with full capos, with other partial capos, and with altered tunings. Usually though we use them to provide more convenient/easier/functional open notes and reduce the number of non-scalar open notes. Oh yeah, and the drones are cool too.
Now where's my hacksaw? Mooh.