One has more than a prayer of separation at an uncontrolled airport. One is fully responsible for separation, and if one can't assure that, one has no business being there in the first place.
The overhead approach is an excellent means of putting one's self in the ideal spot to observe all traffic operations to the runway, and to be seen. It's an excellent way to blend with traffic and set up the landing.
Unless you have a specific need to land on that runway perhaps it's better to let ATC choose?
If you've monitored ATIS, you already know which runway is in use. Likewise, if you're monitoring the traffic you already know this. If you have a different runway in mind, of course, you'll cite the specific runway.
Specifying the runway, even if it's the runway in use, makes your intent and request clear.
If one requests an approach and is cleared and approach without citing the runway, one is cleared to any runway. If one requests an approach to a specific runway and is subsequently cleared as requested, one has a clearance to that runway.
If I approach an airport where landing traffic has a slight tailwind, I may determine I want a different runway. I'll tell ATC what I want. I had a problem several years ago involving a hydraulic failure in a conventional gear airplane. When I arrived at the fields, winds weren't favoring the runway in use. The one I needed was closed, with men and equipment on the runway. I requested the closed runway specifically because it better suited my needs, and the barricades were moved, and the workers and equipment removed.
You could wait to see what's given you, but why not be assertive and specific in your request. After all, you're requesting a specific type of arrival that not everyone will necessarily be using. Why not specify the runway, as well?
But then again you do ask for a specific runway when taking off
Sure. Why not? If I want something other than what's in use, yes. If I want or need that runway. If I'm going to do a normal departure like everyone else that day and plan to use the same runway, then there's no reason to specify. However, the overhead arrival and pattern isn't often what everyone else is using, and if I'm going to do an unusual departure then I'll specify it along with the runway I need.
It's not uncommon to have an unfavorable runway in use; perhaps the weather or winds have changed since they began flying approaches to that runway or since they began filling that pattern; that doesn't mean I want to use that runway for takeoff or landing. At some busy places, my request won't make much difference; JFK isn't going to change the New York area procedures because I say so. l the local airport, on the other hand, often will. Furthermore, everyone may be content flying off with a strong crosswind in their tricycle airplanes, but I might want something more aligned with the wind if I'm in a conventional gear airplane. Ask for what you need, or state it up front. Plain English still works just fine.