Question about no dig gardening

That's great to have that confirmed @MCardiff - it just didn't look willow-y to me!
Lots of plants will tolerate both sun and shade, so you may have to experiment and see what works best. If it's generally windy [same for me] then avoid too many lanky perennials that need lots of support, if you can.
I'd expect your soil to be quite poor under that tree, so I'd have thought some manure or soil conditioner would be fine, but I don't know where you are or your soil type etc. Any organic matter will help the structure, and that's what matters more than anything.
The type of plants you grow will be dictated by your conditions/climate, as well as the general soil health, but it's likely to be fairly dry through summer due to the tree, unless you're in a very wet area. The fringes of the bed, especially the front, might be damper. The more plants you have, the drier it becomes too.
You'll have to assess it all as you go along - some plants will do well, and some might fail. It happens to us all though. The better the prep, the better the chance of success, but it's also worth taking a look at other gardens nearby - beds in the same sort of aspect, and see what's growing well. If you can, ask those neighbours too, especially if you like something, and don't know what it is

 

Improving the soil in that area will encourage your tree to be more vigorous and require more frequent pruning. I struggled with a similar area under a large Acer. Nothing really established well because of the dense root system and dry soil, apart from a shallow rooted grass, Hakonechloa, which you could try, along with some small spring bulbs like Anemone blanda.

 

I’ve grown Hakonechloa in deep dry shade under a tree, alongside Liriope muscari, Uncinia rubra, Berginia Oberture and Aster diveraticus. I gave them some initial tlc re watering but only until they showed signs of been established (new growth).

Just remembered also Hydrangea Limelight, again only watered initially and an occasional can full when we had a prolonged dry period (3 weeks or more)

 

This is Hakonechloa macra aureola under my large Acer @MCardiff. It doesn't need improved soil, is shallow rooted and will cope with dry conditions once established. It's also easy to divide and multiply!

I've also got a Hydrangea nearby @Butterfly6 but not directly under the tree. It's interesting that you're had success with Uncinia rubra in shade, I've got one in full sun. I'll try it in shade next year when I divide it.

 

Very nice garden @Plantminded. You too are obviously very keen on shape and form rather than a riot of colour. I also like the Japanese grass but yours grows better than mine as I live in quite an exposed high altitude area and the grass can get burnt by the wind. The other snag is it is painfully slow and relatively expensive to buy. After 10 years of growing it I now have just about as much as I see in your photo. Any good tips on getting it to bulk up quicker?

 

Thank you @hailbopp. Yes, you're right, I like green and different leaf shapes and forms but there's more colour on the other side of the path which gets more sun! My Hakonechloa is in a sheltered area and the winters are quite mild here. I suspect the temperature and aspect in your garden may be holding it back. Mine has bulked up nicely within three years and is ready for dividing next year. Perhaps you could try moving one to a more sheltered location, nearer the house, to see if that helps. I also have some growing in very shallow soil alongside my steps but it needs extra watering if it's been very dry. Some colour for you :

 

Look @Plantminded don’t say you have got all that grass growing in 3 years. I have been growing it for about 10! I have moved bits about the garden to see if I can find the best spot but where I can still see it. I think I just have to accept that it will look nice for a while and then probably get trashed. A little like some of my Japanese Acers which I also like. You garden is a credit to you and not the easiest as it looks on a fair old slope!

 

Sorry about that @hailbopp! Yes, it's not been in that long. My soil is very sandy here which it seems to like. Another option is to grow it in a tall container where it can show its shape better. Yes, my garden is a challenge, I've been cutting things down a lot this year so that I don't have to use a ladder on sloping borders!

 

Hakonechloa grows really well here @hailbopp. I have H. aurea/aureum, as the variegation's nice, but it loses that if in more shade. It likes plenty of moisture so it's very happy in my garden whether in a raised bed or directly in the ground, and in plenty of sun. Despite creating shelter from wind, my garden's still quite exposed to it, but possibly less altitude than you [we're only around 400 feet here] and probably still more sheltered than your plot. Acers can certainly get a bit shredded on the edges from the wind[not usually the sun!] but mainly just the dissectums. It's always tricky finding the best site in this plot for those.
I expect if you have that grass in a less protected site it'll get shredded quite easily. I stuck a little piece in the front garden where it gets a bit more exposure, and it doesn't do very well there. Perhaps if you can find it a bit of shelter from the worst of the winds, and keep it wetter - maybe create a little area that's almost boggy, that might work?