I would say that mouthful of water "blow back" is just good practice to keep the filter free and clear as much as possible, so that you don't get into a situtation where you need to use the syringe.
The syringe will produce pressure greater than your can with your mouth and would need to be carried in case you really get the filter clogged up.
As was posted above, having good/clear/clean water selection is always key with any filter. But if you have to filter mud water and the filter slows down or clogs up, you will need that syringe.
I also hike in the Sierras and have only had one experience where I need to apply a lot of pressure to backflush my filter in the field after multiple days of use. Basically, I had to almost sit on my "clean" Platypus water back to generate the pressure needed to backflush my filter. Once I did that, I was good to go for the rest of my trip. It was at that point that I started to make a point of doing a little back flushing after each use.
In my case, as a gravity filter, I just squeezed my clean bag to force a little bit of water back through the system and I was fine. Not even a full mouthful of water.
I have used my mouth to "blow back" some water through my filter too, so I know it works with MY setup.
Don't see why it would not work with yours, as it is the same technology inside.
Key things for taking care of the Sawyer filter....store it dry, well as you can.
Before you go to bed, suck the filter dry with your mouth.
Freezing will rupture the plastic hollow tubes inside of it and ruin it....and you really won't know that it happened until you are sick/ill.
True of all filter, but good habits start now.
I make a point of sleeping with my filter in colder weather...just having it wrapping in a small wash cloth that I carry and putting it inside the head of my bivy bag that I sleep in.