If the car is precious to you, don't do it.
Old carbureted cars (not fuel injected) are so unusual to valet workers and so vastly different compared to the modern cars of today that it just isn't a good idea to let any regular valet drive one of these, or attempt to start one.
First, they aren't going to know that they should pump the gas before turning the key, so they will be putting wear and tear on your starter while draining the battery.
Second, the "brake feel" of older cars, especially those with manual brakes, might take a valet by surprise. They are more likely to have a damage incident with these types of cars.
And third, if this happens to be a stick-shift car, most valet workers have poor manual transmission skills to begin with. So if they have to deal with a heavy clutch on an old car—oh boy!
During my 22 months as a valet driver at a busy luxury hotel, out of the more than 10,000 cars I moved, I encountered only two carbureted cars.
What made me different from my co-workers is I was older and had owned older cars during my years (*Slight Boast* including a 1972 Super Sport Chevy Nova with a 350 and a 4 speed). So I was confident with the 1960's Firebird I had to park that one time. But I was not as confident with the 1956 Rolls...
It was the day Hurricane Sandy blew into Philadelphia. The Rolls, along with a number of other vehicles, was in danger of being flooded... because our secure parking area was filling up with water from the storm.
It was an emergency situation.
The 1956 Rolls Royce was in pristine condition. And I knew it was up to me to move it. However, this was a car which was completely foreign to me. I didn't know if it might be fuel injected (possible). So I hopped in and turned the key without pumping the gas. The engine turned, but it didn't fire. So then I pumped the gas a few times and turned the key once more.
The key broke right off in the ignition!
And it didn't start. The ignition was now in the "on" position, so the battery was now being drained. Meanwhile, water was rising all around the car!
Well, I had to get help. In the end, everything worked out okay with all the cars. It was a bit of an embarrassing situation for me though. And after that I refused to have anything further to do with that car, because that was a tricky car and I had no knowledge of its various nuances.
It's going to be a similar situation for any 19 year old valet who has to start-up a cold carbureted engine. Your expectations are just too high if you think there will be a valet on duty who is familiar enough with old cars.