Let's look at "pool store chems' vs. BBB BBB is not some 'magic method', it's just good pool maintenance!
Chlorine is chlorine once in the water. There are several kinds, however, and each has various effects on your water balance
Trichlor as used in puck and dichlor, sometimes sold as 'shock' are the stabilized chlorines. Both add CYA to the water and both will eventually lead to an overstabilzied pool. They are also acidic and will cause your pH and TA to drop over time.
Untabilized chlorine does not raise CYA. There are three of them, Sodiium Hypochlorite, Calcium Hypochlorite, and Lithium Hypochlorite. They are close to pH neutral and less impact on pH and non on TA. You might need to add acid when adding them because they are not acidic like the stabilized chlorines.
Sodium Hypo is the least expensive and has the fewest "side effects". It is called liquid chlorine. It is also called unscented laundry bleach (aka Clorox and other brands). These are one and the same thing! Only difference is the concentrations available so the dosage is different to achieve the same FC level. Pool chlorine comes in 12.5%, 10%, and 6% while bleach comes in 6% (ultra) and 5.25% (regular) strength. Other than the strength there is NO DIFFERENCE!
Calcium hypo is commonly sold as shock in 1 lb bags but also sold in big drums for chlorination. It will cause your calcium hardness to rise and can make the water cloudy.
Lithium hypo has very few side effects and is fast dissolving but it is VERY EXPEN$IVE to use. If you can afford it, go for it!
There is no such thing as 'shock'. Shock means using an unstabilized chlorine to raise your FC levels high enough to break down chloramines or kill algae. Shock is a verb (something you do to a pool), no a noun (a special product). Dichlor is sometimes sold as a shock. This is a sure fire recipe for an overstabilized pool!
Total Alkalinity Increaser vs Baking Soda--one and the same. NO DIFFERENCE at all! Both sodium bicarbonate and both made by Church and Dwight (Arm and Hammer!)
pH increaser vs. Arm and Hammer Super Washing Sods--one and the same, both are sodium carbonate and both made by Church and Dwight once again. 20 mule team borax is a better choice for raising pH BTW! (and yes borax is used in swimming pools. You can buy several brands such as Bioguard Optimizer, Proteam Supreme, Poollife Endure, Guardex Enhance, Mineral springs pool salt mix is mostly salt and borax as is Salt Water magic.)
Muriatic Acid or dry acid (sodium bisulfate)--no substitutes here, we recommend the muriatic because it does not add sulfates to the water.
CYA (cyanuric acid)--no substitutes here either....If you need it buy it and put it in.
A GOOD test kit---this is a necessity!
All BBB is really is just good pool maintenance, just like the professionals use to care for commercial pools.
I know becasue I am one!