Should my Omega wind?

Maybe the reversing wheel is worn out and needs to be replaced. Most auto Omega's from this era I have ever handled don't wind manually because of this.

Yes. You should hear it winding when you turn the crown clockwise in the position closest to the case. If you don't hear it winding, then it's likely that whoever serviced this didn't do the job properly.

If you took this to a jewelry store, did they have a watchmaker on staff? If not, they are likely sending it out to someone and doubling what the watchmaker doing to work is charging them. Always best to find a watchmaker and work with them directly, rather than going through a jewelry store.

Cheers, Al

The watchmaker through jewelry store is not uncommon- my watchmaker with whom I have been friends for 20 years stopped taking private clients about 10 years ago- he works out of his house and got tired of people ringing his door bell at 7pm to get a battery changed.
He only takes work through the local jewelry stores now.

Yes, it is common. There are some potential drawbacks with getting work done through a store like this - trade work as it's called...

Often these stores are looking for the lowest priced watchmaker, so they get a bigger margin on the repairs. The watchmaker is often put under unreasonable time and price pressures, and as a result the work suffers.

One is that you are relying on someone at the store, who may not know anything about watches, relaying information about a potential problem to the watchmaker when the watch is returned. This can lead to multiple trips back and forth to sort out a problem.

Sure it can work, but generally working directly with a watchmaker is the best way, so you have clear communication with the person who is actually doing the work.

Just to clarify- I’m not saying going through a jewelry store is a good way of doing it, but in some places finding a good independent watchmaker is difficult and unless you can get past the jeweler to find who their watchmaker is, you are stuck unless you can ship it to a watchmaker out of town - of which there are many who do business that way including many on this forum).
I was lucky enough to get in with my guy when he was still taking private clients so I’m grandfathered- but when he retires, there will a void in my area as there aren’t any others in the immediate area that I know of.

I think in some (many?) cases, you don't want the guy the store is using. I know your situation is different, but often the guys doing trade work are not doing the work properly, and will not do things differently when they work for someone directly. Bad habits don't tend to suddenly go away just because the person doing the work is dealing with you directly.

In the industry, trade work is considered sort of the lowest form of work in many ways. Many watchmakers simply refuse to do it, as it either requires you to lower your standards, or lose your shirt doing the job properly and not getting paid for it. Not saying all situations are like this, but that's often the reality. As I've said before the watchmaker that was giving me advice when I started out did trade work and hated it, and complained continuously of the crappy things he was forced to do to make a living. But he hated working with "picky collectors" so refused to take on work directly from people.

The availability of watchmakers is only going to get worse over time, as more watchmakers retire or die off then are replacing them. I think finding a "local" watchmaker is going to be a thing of the past, unless you live in an area with quite a large population. Shipping watches for service is really going to be the norm for most, unless you are lucky...

Cheers, Al