They didn't find him by going door to door, though, and actually that was a very dangerous thing to do. It could have easily become a hostage situation if he snuck into a home and was as armed as they said. Telling everyone to shelter in place made it easier for people not to be missed, as opposed to "Oh God, Helen didn't come in to work today and she lives right near where this happened! Maybe something went wrong!"
The airport already has security, right? I would think that this guy would stand out, and keeping the airports open BUT BEING ON THE LOOKOUT FOR THIS GUY would have been pretty smart. I am guessing that's why it was allowed for a bit (you can be certain the Feds were checking passengers against the video/stills).
Bus routes are also pretty easy to follow, and since this was a deviation in their plans, it would be narrowed down to people purchasing tickets after the gunfight.
None of these are superb, and certainly privacy is still invaded, but yeah they are viable alternatives to shutting down a city and jailing all the law-abiders to see who the criminals are (anyone going outside, obviously, because criminals don't know how to blend or hide).