10 sessions for me is a short campaign. Most any system would work for a 10 session campaign. I have done 3 to 4 sessions as a short campaign and I have done 10 to 12. 10 Sessions has time to develop story and some character development. I think players could accomplish one or two short term goals but nothing epic. I probably am going to sound like a broken record but I have two primary systems for short campaigns like this.
Savage Worlds is one of my go-to systems, although it took some time to get there. Although it has its own nuances that one has to get used to, there are not a ton and they are very consistent and they make sense. Characters are easy to make and one can get started right away. It has a slow and subtle progression character development, but I would not expect more than a few advancements in 10 sessions, maybe just enough to get from Novice to Seasoned.
If you like Lovecraftian horror or pulp action, Call of Cthulhu really suits itself to short campaigns. I have run many short campaigns in most every edition of CoC. As a horror game, the system and the setting combined tends itself towards shorter campaigns and one-shots simple because of the insanity factor. It also does not have a structured character progression system, so a person really does not feel like advancement changes or differentiates the character long term.
Finally, in my experience, the d6 system is also best in short campaigns. Other more or less a dead system because the company no long supports it and dumped the entire system on the web as an open system, in my opinion it is still viable. I ran it for multi-year campaigns in d6 but what I found was that most characters gravitate to the same kinds of skills and there is very little differentiation between characters after too long. I was also told by one of the original writers for the system that it was truly meant for short campaigns. If you have not tried it, I highly recommend looking into it.
I am finding it harder and harder to run epics campaigns like I used to, so 10 to 12 session campaigns are more common. They can be fun too, as long as you balance story driven elements with character driven elements, so that the players don’t feel railroaded.