Here's my thoughts on the garden...
Your lawn is very striking and beautiful. I would hesitate before planning too many elements such as lots of shrubs and flower borders.. The beauty in this garden is it's scale and distinctive, captivating simplicity - it has a minimalist and formal garden feel to it, which is very calming and soothing. Developing these themes will keep the low maintenance you want, whilst preserving the unique and tranquil character of the space.
A few ideas spring to mind - rather different perhaps to other suggestions you've had, so it might not be on your wavelength, but here goes...
On a practical note - don't paint the fence! It will detract from the lawn and will take hours of work in upkeep - and it won't look as good as the natural wood. Painting rarely helps to preserve it in my experience, as mostly it's the posts that eventually need replacing not the panels. (If you do, use a wood stain close to the natural colour - it will need less upkeep).
Planting - Because the garden has a more formal vibe, I would try to mirror the tree planting that is on the right hand side onto the left. The garden scale needs some big structural plants and trees are perfect for this (also very low maintenance). Mirroring the tree-planting will give balance to the lawn area.
Water feature - Two giant urns on the patio - large enough to match the scale of the garden - one each side of the steps spaced evenly; with solar powered water fountains. Intersperse with fewer patio pots but of larger proportions than you have at present - perhaps planted with some topiary, or in some way striking in features.
Decking. I know you said sunny, but the end of the garden extending out from the cabin area seems a perfect spot - along the back length of the garden and framed with a pergola or series of archways. It would add interest to the end of the garden, provide a cool place to visit and enable some climbers to draw the eye from the patio.
In front of this I would plant a row of silver birches - giving a distinctive vista in summer and winter, but not hiding the decking too densely.
I think this would develop into a striking garden, very low maintenance and keep the elements of simplicity and scale.