doesn't this mean that method is basically synchronous
No. It's asynchronous. You're probably thinking of sequential (progressing from one thing to the next), not synchronous (blocking the current thread). An await will pause the method (sequentially) but not block the thread (asynchronously). For more information, see my async intro.
without the async modifier
While you could elide the async/await keywords, I would recommend that you do not. This is because // some more code here may throw an exception. I cover this and other considerations in my blog post on eliding async and await.
and the suffix Async?
No, that suffix is appropriate for any method that returns an awaitable (e.g., Task). So, even if you elide the async and await, it's still returning a task that should be awaited, so it should still have the Async suffix.
You can think of it this way: the Async suffix is part of the API interface. The async keyword is an implementation detail. They often go together, but not always.