What questions didn't they answer regarding their own dawn search?
They were alone during their own search as they saw no-one else searching - probably because everyone else who had been searching had long since gone home to bed. Kate was not claiming no-one else searched -she was describing how she felt at that particular time (when they saw no-one else searching for the whole time they were out) - which was completely alone and in despair.
I assume they searched between 6am and 7am or thereabouts. At the same time, Kate's phone was very busy, mostly texting. The last time her phone was used previous to this was 4.55am;
6.02.08 am Mum mob calls Kate (0 seconds) SMS
6.04.11 am Kate calls Mum mob (0 seconds) SMS
6.05.29 am Kate calls Amanda home (7.02 minutes)
6.08.17 am Jon Corner calls Kate (0 seconds) SMS
6.18.17 am Kate calls Jon Corner (0 seconds) SMS
6.34.53 am ... calls Kate (0 seconds) SMS
6.35.23 am Kate calls ... (0 seconds) SMS
6.39.38 am Kate calls Jon Corner (0 seconds) SMS
6.47.42 am Jon Corner calls Kate (0 seconds) SMS
6.47.54 am Jon Corner calls Kate (0 seconds) SMS
6.48.05 am Jon Corner calls Kate (0 seconds) SMS
6.59.12 am Jon Corner calls Kate (0 seconds) SMS
6.59.44 am Kate calls Jon Corner (0 seconds) SMS
07.00 am GNR Officer P. J. F. N. saw the McCann couple at about 07.00 alone in the street next to the site where they were stationed.
7.06.06 am Sue and Brian Healy call Gerry (01 seconds) SMS
7.09.04 am John McCann calls Gerry (3.26 minutes)
7.15.19 am Gerry calls Angela Morado UK Consulate (4.51 minutes)
7.23.20 am ... calls Gerry (5.53 minutes)
7.41.14 am Jill mob calls Kate (3.30 minutes)