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Increasing noise (grain) is going to be an issue. You cant get detail where none exists due to the exposure.
Try LEVELS instead of brightness/contrast.
Then add CURVES
Another thing to try... Place the clip on the timeline. Now, Copy it to a higher track. Turn off top track. Now, apply levels and brighten it up but make it overbright. Now ad a Blur to the clip... Not too blurry, but a bit.
Now, turn on top track and lower opacity of top track revealing the overbright blured track below. This should be able to get it brighter without adding grain. You will need to tweak it, but I've gotten acceptable results with old footage like that.
For similar, I will start with Levels and adjust. I follow up with Highlight & Shadow (take that off of Auto, and maybe look at about 20 - 35%), then go back to Levels, if necessary. Curves is also very powerful, and might well do all that you require.
When one encounters the "grain," that Craig mentions, I find that Neat Video does a good job, at reducing the "grain," but will increase the processing time. I will then add a touch of Unsharp Mask, as the very last Effect.