Is it common for such basic TTL gate chips as 74LS32, '08, and '86 to be broken?
I used 74LS logic gates on several projects before 74HC(T) came along.
The 74LS outputs were damaged easily by accidental shorts or overloads. Such shorts came about through careless use of 'scope/DMM probes, bad wiring in wire-wrapped boards or lash-ups and so on. I saw ICs with visible heat damage or even sections of package cracked off by it.
In comparison, outputs for the modern logic gate families and CPLDs/FPGAs I've used since the late 90's or so are pretty tough. Although an accidental short still makes my heart jump, I've not yet seen an IC damaged by it.
So this needs you to look into how you are using these 74LS ICs. Review your circuit and implementation and check that you have no overloads, outputs connected to outputs and so forth. Look into your debugging practices.
If the 74LS ICs are operating within their published specifications at all times, including during power-up, power-down and unpowered, then you need to consider the quality of the supplied parts.