Synthetic oils are slipperier than conventional oils. That's great for lubrication and wear reduction, but tends to leak more past old seals, gaskets, and worn rings.
Syn oil being slipperier makes it easier for it to slip past old seals, gaskets, & worn rings in worn cylinders. Thus syn is more likely to leak, burn, or use/lose oil, especially in an old car that's high miles. Even moreso if that old HM car had used conventional oil its entire life because it'd have more wear.
Back in the day, the ideal time to switch to syn was around 30K miles, but 50K was pretty good too. IME 75K miles was still safe to switch to syn if the car had been well maintained.
The part the service guy said about sportcars probably meant only cars driven very hard needed synthetic. I agree that was true back then. However, he should have also included vehicles that haul or tow heavy loads.
I also agree with what was said in post 48 about oil pressure. I experienced same when I switched a 99 Jeep 4L with 30K miles from 10w30 conv to 10w30 Mobil One. Low oil pressure at idle. I resolved that by switching to Mobil One 5w40. Then ideal oil pressure.
If/when I switch a HM car from conv to syn, I choose a syn that's thick for its grade, or go up one grade. Then oil pressure still good and hopefully no leaks or oil burning.
I agree with what the dealership service guy told you. He didn't want to risk starting/causing an oil leak, low oil pressure at idle, or burning oil. If any of those problems began after he changed your oil, you'd come back and complain, which would make it his problem.
In current times, a lot of modern engines are less robust, less durable and need syn oil just to survive normal driving. i.e. - A lot of modern engines are whiny snowflakes. The good thing is that the modern engines that need syn have it in them their entire lives. So there are no changeover concerns.
The engine tech has changed for better and worse. Mostly worse after 2005, IMO. Oil tech has improved, but that is somewhat cancelled by some modern engines being designed more fragile than used to be.