I couldn't disagree more with your tranny shop guy.
Petrol based ATF is an insulator and does not like to release heat. Synthetic oil releases heat at nearly the rate of water. The other advantage is that petrol based ATF will literally "burn"...... it's an organic material. Synthetic will not burn and won't break down until ~600* (long after the seals fail).
If you switch to synthetic, be sure to get all of the ATF out of the converter. There is an easy way to accomplish this. Most of the 4L60E trans hold about 9 qts (4 in the oil pan and 5 in the converter). Start out by buying 10-12 qts of synth ATF, a new filter and trans pan gskt. Pull the pan (clean it), replace the filter. Replace the pan and fill the pan until the dipstick says full (do not start engine). Remove the return cooler line from the radiator. Put a rubber hose on the line and run the hose into a clear container that has "1 qt" measured and marked on it. Start the engine and let it idle until 1 qt of ATF flows into the container (around 10 seconds). Shut-off the engine and add one qt of synthetic ATF to the trans. Dump the fluid from the clear container. Repeat this process until you see a change in the color of the fluid pouring into the container. This should take the remainder of your 10-qts. If you have an over-sized oil pan it may take more. You are better off buying more ATF than you need, you can always return what you don't use.
If your vehicle is just a daily driver, you should not have to worry about changing the ATF for another 100K miles. If you tow moderately, 60K miles. If you tow heavily, 40K miles. The bottom line is, synthetic will never burn but it can get contaminated by moving parts wear. The heavier that you tow (race or mountain drive), the more wear that you will experience.
Hope this helps.
This is my trans cooler set-up:
This is my engine and trans temps towing 4,000# in 90* temps.