I don't know about the details of the fence, but after whatever is done with that: I think planting a vine to train up the inside and over the top would be a great finishing touch! Probably help with the sound insulation, too... with the exception of those admiring comments you -want- to hear that it would be fun to catch from your hidden location gardening on the other side.

Great idea! The vine can also help with the privacy issue!

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Your first idea is the best, as caulking, the holes with boards is the safest option. My fence had just as many gaps, and I didn't like it. I thought about building a new fence, but it is too expensive. After analyzing all the options, I decided to buy special garden screens here, https://www.luxuryscreens.co.u..., and install them so they would close the gaps. My idea worked perfectly, and now people outside can not see what is happening in my yard.