I'm afraid it would be pointless having Pyracantha, for the reason I gave earlier. You would constantly be trimming off the flowers and then the berries, and would just have the green foliage of that instead of the privet. I can't see any kind of cherry being any more suitable. If you want something nicer to look at, you'd really have to grow plants in front of it, but you'd need to leave a space to access the hedge for maintenance.
It's a place where beautiful isn't enough of a word....
I live in west central Scotland - not where that photo is...
You can certainly cut the privet down the middle or down to the ground without killing it. We inherited a very old, leggy and gappy privet boundary with the churchyard next door. Having obtained the necessary permissions we cut the hedge right down to the ground taking out many dead bits and even more litter, mulched the stumps with compost and gave it a good watering. Within weeks it had greened up and started to grow again. Within a couple of years we had a much better hedge and the following year it had reached the height we wanted. The hedge is now dense and a very efficient boundary.
THe downside is, as has been pointed out, that privet is very vigorous and it takes a lot of goodness out of the soil. It is not impossible to grow something next to it but I would not plant anything as close as you are suggesting if you want it to grow properly and if you want to be able to trim the hedge (and an untrimmed privet is not usually a lovely thing in a hedge). I think that your options are either to learn to love your slimmed-down hedge and keep it trimmed or have it out and plant something else instead. With our privet we have quite a wide border in front of it so have been able to plant this up with shrubs and perennials, but not right up to the hedge.
If the hedge runs alongside your patio, have you considered placing some pots on the patio and planting them up with something you like? Not having seen a photo of your garden, is there any scope for creating a raised bed a short distance in front of the rest of the hedge and planting that up? Neither of these options would have a negative effect on the hedge and they will provide your chosen plants with their own supply of water and nutrients. You'll need to think about access for cutting the hedge though and you'll need to give your pots and raised bed attention too.
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If you want something nicer to look at, you'd really have to grow plants in front of it, but you'd need to leave a space to access the hedge for maintenance.
I live in west central Scotland - not where that photo is...