I think we need to separate the sociological phenomenon from the physical object. I fully agree that logo’d T-shirts were around before the 1970s (I recall one radio station giving them away as promotions, but they were novelties). To me, the interesting thing is the boom.
I’ve seen T-shirts and sweat shirts in old movies (1940s/50s) emblazoned with the name of a gym, or (similarly) schools, but these acted almost as uniforms at a time when uniforms were more common and their use more specific. Today, only certain categories wear uniforms [e.g. food servers (from good humor men to tony waiters); service workers (custodians, utility workers, doormen, mechanics); medical staff) even if it isn’t strictly necessary to their duties, but the same trades often wear street clothes or something akin to it (e.g. some waitresses may wear white shirt and black slacks that wouldn’t be out of place on the street). In the 1950s, defined ‘expected’ dress was the rule: IBM famously dictated the style and color of a sales rep’s shirt. I was thrown off some tennis courts in the 1970s because my stylish Jimmy Connor tennis outfit had trademark red and blue stripes on the pocket slit and across the chest–they weren’t “proper tennis whites”
As late as the 1970s wearing a baseball cap suggested that you actually intended to play baseball in the imminent future (some kids wore them all the time, but it was often a personal quirk, and pick-up games were common anyway). It wasn’t a taboo --as evidenced by the flurry of formal rules against, e.g. wearing them to school, once that began to become common. It would simply have seemed as out of place as a diver wearing a facemask on top of his head during his daily errands.
I don’t want to overgeneralise. Caps were quite common in some settings. Farmers (and many people in rural communities) wore them, probably because they worked out in the elements. Truckers wore them, possibly as an evolution from the delivery person’s cap that was common in the 50s/60s, but in both cases, they were distinctive traits. Today, the guy in the hardware store with a baseball cap might be a lawyer or banker. you can hardly walk down the street today without seeing many baseball caps, but if you watch a 70s movie or TV show, they’re fairly uncommon.
I think that we can date the “fad” of commercial logos on T-shirts pretty effectively by looking at the complaints against it. T-shirts with slogans and noncommercial logos (e.g. peace sign, smiley or rock group) were common in the 60s, but I can recall there being significant grumbling in the 70s when they were usurped by commercial logos. (A few companies, like Coke, whose icons were cultural phenomena of their own, actually snuck in under the wire on this one, but it was roundly criticized in the overall backlash some years later.)
Fairly or not, I’m going to implicate the US Bicentennial. I don’t think it’s a complete coincidence that the shift occurred ca. 1976. Starting in about '73-74, many small or impromptu businesses sprang up to peddle T-shirts commemorating various local celebrations of historical events (e.g. not the original event but its patriotic "festival). Before that you usually GOT logo’d T-shirts; after, say, 1976, it became more likely that you BOUGHT them. I remember thinking it was odd to pay for them. Prior to 1974, the practice was limited to tourist sites.
This flows into the related (and even more loudly decried) practice of putting designer labels on clothing (e.g. designer jeans and Izod shirts) later in the 70s.