Old privet hedge, remove or repair?

I agree with the others, it doesn't look bad at all. The bare bits near the bottom at the inside aren't really a privacy issue because of the wall (which is pretty much identical to ours!).
If you cut the top a bit narrower and maybe a bit lower it should allow more light to the lower part and stimulate growth, then you can train some new growth across the gaps. Maybe also shorten some of those long straight vertical branches on the inside - they will shoot from just below where you cut which will thicken it up at that height.