24-105 f/4 L worth 2 Tamron EF-S lenses?

I am contemplating on selling/trading my Tamron 17-50 f/2.8 (non-VC) and my Tamron 18-270 for the Canon 24-105 f/4 L to be used with my 7D.

I primarily take portraits (people shots) and found the 30-135mm range to be very ideal for this purposed on a crop. My primary frustrations with the Tamron lenses is not the IQ, but the slow AF/hunting when focusing from near to far or vice versa, since I take a lot of portraits of moving kids indoors and outdoors.

I have a 580 EX II which I hope will work well with f/4. I have the 10-22mm for the wide angle range and also have the really old EF 70-210mm f/4 which can probably help with the telephoto range that I would miss with the Tamron.

Based on this, would the 24-105 f/4 be a significant improvement in terms of AF speed/hunting indoors?

IQ (sharpness, saturation & pop) and AF speed are usually the only things that concern me. I care little about body build, weather proofing, etc.