I've no idea if that is what happened in practice but I don't see irrigation on its own leading to multiple tees when you consider that for centuries players of varying abilities and lengths have used the same tees.
Niall
Well, I've played something like 65 rounds on links courses in Scotland. (one with you, thank you very much) And even as a low-trajectory, short-hitting old guy, I have rarely felt overwhelmed by the course. Exceptions are Royal Dornoch and Jubilee in St. Andrews for different reasons, I"ll address them later.
Here in the US, however, it's happened a lot more often.
You are probably right that irrigation alone isn't the driver, but I can assure you that in the middle of the country, wall-to-wall irrigation has had a HUGE impact.
The other thing that happened at the same time was the perceived need to have courses that "test" the best players.
The nine-hole course I grew up on in northern Minnesota 65 years ago, when I was getting serious about golf, had two sets of tees. One barely over 3,000 yards and one about 2,800.
No men played the forward tees, no matter how old or short hitting. As far as I can remember no women played the longer ones.
In the spring, it was cold and the grass didn't get really thick, so even though we mowed the bluegrass at nearly an inch, it wasn't too bad. I don't remember much, but during June it had to start getting pretty slow, but by July the lack of irrigation meant fairways dried out and even with balata balls and persimmon woods, the 400-yard first hole was often reachable for me with a traditional blade 8 iron. And I was never a long hitter or good iron player.
About 20 years ago I was back and playing with a modern ball and driver, couldn't get there with two woods.
About the same time as full-coverage fairway irrigation coming to most of the courses I played (1970s and 1980s), Courses started getting longer...often times by adding "back" tees.
New courses not only had to have tees at almost 7,000 yards to be considered serious, forced carries suddenly started cropping up.
So, by the 1990s and the Multi-layer ball revolution, the die was cast.
BTW- my old home course is an perfect object lesson Niall. Like the courses in Scotland you talk about being played off two or three tees by all abilities, the original nine is still intact, and has a 206-yard par three, but a 145-yard par three off an elevated tee, a 254-yard par and a par five that's only 415. So despite being over 3,000 yards, a third of the holes are easily reachable in regulation.