But if that was true, then hiring a company to "manufacture 1,000 shirts" would be a service, because "manufacture 1,000 shirts" can be put in the place of X.
Well yes. The service here being the manufacturing under contract. Now if you purchased a 1000 of their own brand of shirts or preproduction general merchandise then that is you buying their product.
The service being provided here is their manufacturing capacity being contracted to you based on your order. Technically your output is still a product but what you are paying for is the manufacture. Its one of those mixed cases I guess.
Its still manufacturing capacity and they do produce goods. But what they sell is the capability to have those goods created on demand. The shirt would never be produced if you did not order them to make it.