Interesting question, with pros and cons on both sides. Bottom line for me: skip the worry with the vendor you have and just go to OSH Park to keep it simple.
Here's the summary as I see it:
Pros (reasons to use this board house):
- save time
- avoid making errors if you are unfamiliar with generating gerber files
- fabricators have full access to your design (so they can adjust it as needed for their particular manufacturing process)
Cons (reasons to use a different board house):
- fabricators have full access to your design (they could introduce errors accidentally, or they could steal your work, though that's unlikely)
- you don't get to make choices about the way your design gets manufactured
- you don't get the experience of making gerber files and learning about the process
Overall, I've never heard of a vendor who refused to accept gerber files, so it does sound like a red flag to me. If this is a hobby project, not something you will make a million copies of or something you want to make a lot of money from, then it's probably not a big deal. However, I would highly recommend OSH Park for hobby PCB fab, as they have excellent customer service to help you through the output file generation process. Their prices also can't be beat for low volumes, and they are particularly happy to work with students.
Eventually you will want to find a PCB vendor that you like to work with, maybe someone local to you, and build a relationship with them. Knowing your vendor's particular process will help you a lot with design choices, such as how small you can make your trace widths and spacing, how small you can make your soldermask web, what drill sizes to use, etc. See if you can get a tour of a PCB fab shop so you can understand how the processes work -- it's really fascinating!
Communicating with your vendor is important so you can make sure your design will turn out the way you expect when it's made into a real physical object, not just the idealized image you have on your screen. Vendors really appreciate PCB designers who are willing to work with their requirements, and once you get to know them they will often go the extra mile for you to make sure your boards come out well.
Best of luck with your project!