Perhaps some numbers might help alleviate your husband's concerns -
8.25% laundry bleach typically has a % available chlorine of ~ 7.62%. That is, bleach is normally sold by weight % of sodium hypochlorite (NaOCl) so if you convert it to the volume % of available chlorine (Cl2), it makes it easier to compare what we normally measure in pools as free chlorine. Most clothes washers (at least for my Maytag Bravo XL) limit the amount of bleach added to 3/4 cup max and the water volume of my typical white clothes is probably around 10-15gal (can be more or less as the machine has an automatic water level system determined by the weight of clothes in the machine). But if we use those numbers we get the following -
0.75 cup * (8 oz/cup) / (15 gal * 128 oz/gal) = 0.3125 %
(0.3125% * 7.62%) * 1,000,000 = 238ppm available chloriine
And that 238ppm is unstabilized chlorine which means it's all hypochlorite (OCl-). Normal swimming pools will have, on average, about 5ppm free chlorine which is stabilized meaning that most of the chlorine is bound to the cyanuric acid molecule and held in reserve. Only a very small fraction (~ 5% or so) of that 5ppm exists as either hypochlorous acid or hypochlorite anion (those are the two active chlorine species that sanitize and oxidize).
So you can see that the amount of chlorine in bleach used to clean clothes is hundreds of times more concentrated than what is used to disinfect swimming pools. Now that's not to say that a chlorinated swimming pool will not degrade clothing. But what people normally associate with "bad chlorine pools" are the types of pools managed in commercial facilities (like the YMCA, etc) where they are not allowed to use stabilizer with chlorine and thus have pool water with much more harsh levels of active chlorine in the water. In places like those, a typically bathing suit won't last more than a season or even less. The methods we advocate create pool water that is very comfortable to swim in and not harsh at all.