First of all, I'm not mobile and I had my shop built on the same lot as my home, so my situation may not apply for everyone (especially mobile guys who need to maximize space in their rig). I personally have a cheap shop vac brand vacuum with a mesh filter that I ONLY use when I need to suck up liquids (I don't like using my ridged because I've had the same issue with it spitting the water out of the rear exhaust).
I always keep the mesh filter in that shop vac that way it is always ready. If I have to heavily saturated an area of carpet because I need to remove a stain or spill, or remove water from a flooded basement I know it's ready.
For 99% of the carpets I clean I use my rigid to pre-vacuum with a brush and crevice attachment, then I use d101 APC diluted 15:1 with distilled water to clean the carpets and a terry cloth from costco to mop up, these are the only towels I wash prior to using due to insane linting but once the initial lint has been removed in the dryer they are truly the best cotton terry cloth towels I've found. I'm not a huge fan of using microfiber when cleaning interior carpet but I do use them on dark color carpet.
Once I'm done with the rest of my detail I re-vacuum all of the carpets with my rigid vacuum. For me, the dirt that transfers onto my towel when I'm mopping it up is the most important part of the process when working without an extractor. The second most important part when cleaning carpets without a hot water extractor is not over saturating, which could lead to growth of mold and bacteria and will leave the carpet crunchy do the apc being left in the carpet fibers.
One thing that I have changed in the past year is my actually dilution of d101. I've always used distilled water with products that are primarily used on paint. But I have an extra bottle of d101 that I keep in my shop that is not only diluted with distilled water but is diluted at a 15:1 ratio rather than 10:1. Between the use of distilled water and the lighter mixture it has truly made a difference in the way the carpet looks and feels once it's dry. I can't even image using so much cleaner on carpet that I could vacuum it up without it evaporating somewhere in the hose.