Alkalinity is a measure for the water's resistance against pH dropping when an acid is added. It is more or less the sum of all substances in the water that can accept hydrogen ions (H
+) minus the number of hydrogen ions itself. pH is basically a measure for the number of hydrogen ions - low pH means many hydrogen ions, high pH means fewer of them. If you add an acid, then you basically add more hydrogen ions, but if there are substances in the water that can accept ("buffer") these additional hydrogen ions, then the pH won't change much.
High TA itself doesn't pull pH up. The point is which substance is used to build up a buffer system in the water. The traditional buffer system in pools is the carbonate buffer by adding bicarb to the pool. This is the Carbonate Alkalinity (CA) part of Total Alkalinity (TA). That means you force more carbon dioxide to be dissolved in the water than there would be if just letting the water in equilibrium with atmospheric CO2. Like a bottle of soda water. If you open the soda bottle and the little gas volume from the top of the bottle can start exchanging molecules with the atmosphere, all the CO2 in the water will think "Hey, the guys above us are leaving the bottle, let's follow them!", and CO2 will out-gas from the water. And in the process the pH rises until you reach a pH where the same number of CO2 leaves the water as re-enters the water from the atmosphere - the water reaches equilibrium.
CYA can also accept hydrogen ions, and has at pool pH very comparable buffering capability as the carbonate buffer - with the advantage that you don't create carbonated water by adding it. The CYA-Alkalinity part of TA doesn't create CO2 out-gassing and doesn't contribute to the dreaded pH-drift.
That's part of the reason why TFP recommends higher CYA for SWGs - it increases TA without increasing pH-drift and without increasing the CSI (only the CA-part of TA adds to the CSI), and therefore helps to prevent scaling in the SWG-cell.
Low TA doesn't by itself pull pH down, but it means that a small acid addition will create a relatively large pH-drop. That's why it is important to have enough TA when using acidic forms of chlorine like Trichlor or Dichlor to prevent pH from crashing way down. For non-acidic chlorination methods (bleach, cal-hypo or SWG), less TA is sufficient to prevent pH from crashing. With Trichlor it is actually desirable to have enough CO2 outgassing to compensate the pH-drop due to the Trichlor - with a low TA, pH will drop very quickly when adding Trichlor tabs. That's probably where the perception that low TA pulls pH down comes from.
If you really want to dive into the deep end, then I'd recommend this post:
Chem Geek's Definition of TA