Most jets have been capable of doing landing automatically for decades. This is allowed only on certified planes and runways and actually required for poor visibility situations. ILS CAT IIIc has no decision height or run way visibility requirements but no airport actually supports that yet. The main reason for this is that taxiing blind is kind of tricky. Landing and stopping the plane on the other hand is a solved problem. CAT IIIb landings are fully automatic.
The plane approaches, flares, and lands and keeps the plane on centerline (in some cases). Most (if not all) currently flying 737s are capable of this. The question is of course if this particular airport was ready for that (if not, I'm not sure if the autopilot would allow this) and whether the pilots were certified for this.
Cat III can be done with no DH, but you still require some RVR>0 (flare is done by the autoland system). Only IIIc could be done in theory with a blocked windscreen like in this case, but that's still not in standard operation anywhere that I know of.
All available categories have a decision height > 0ft: https://en.wikipedia.org/wiki/Instrument_landing_system#ILS_...