Menelaos, I think a 'grove' of small trees (15-20 feet as wide as tall) would work beautifully. At least 3 trees...they grow together well. Install crushed gravel (or pea gravel I like compact able crushed gravel no larger than 3/8inch, 4" thick). This would be an extension of your patio because that 'lawn' simply isn't working for you.
I'll be back to give you suggestions for trees, but first I'd like to change your perspective about trees a little bit...
First, all plants flower. There is not a tree that does not. When they flower to attract pollinators (bees) and those flowers get 'pollinated' they make seed usually in the form of berries, fruits. Most floriferous trees produce fruit that has no chance of making to the ground because the birds eat it faster than it ripens. Serviceberry. I am still working on trying to find if this wonderful tree would do well in Greece. I've listed a few other patio trees for you to consider as well as pictures of alternative ways to make a shady yummy spot in your backyard.
There are some wonderful 'vase shaped' patio trees. Serviceberry is but one. They are called patio trees for a reason; they are in 'scale' with us humans. They have close up aesthetic beauty such as bark or sculptural branching. They will have flowers, delightful to sit beneath on a spring afternoon with soft flower petals falling all around you.
The shade is perfect, thus another aspect of 'patio tree'. Dappled shade not total shade allows more breezes through and you won't need a torch to read by.
"Groves" are one of the things I put in my designs whenever possible. Gravel 'flooring', the trunks and branches make 'walls' and the canopy the 'ceiling'...this takes a good 5 to 7 years to get the maturity. A big tree with dense shade about 20 years.
Are you allergic to bees? They would be there UP in the canopy not bothering anyone. Also important to know is what resources you have for nurseries. Wholesale is best but you'd need to go through a contractor with a resale license. That makes sense here in the States but I've no idea what kind of nurseries you have.
If you have resources nearby please send me a list of their available plants/trees. Yes, they will have a list of all the plants that do the best in your locality. This will help whittle the plant list down considerably.
There is another alternative; a lovely patio setting with potted plants/trees/palms, an easy to maintain floor such as gravel or more pavers/flagstone to match your original patio. That hard edge between your flagstone and the rest of your yard was unfortunate. The idea, basically, is to transition from architectural to au naturale as you walk away from your home.
It would be nice to soften that edge and transition by making a flagstone pathway into your 'grove' where a small area of flagstone paving for the Adirondack chairs a little table for your Raki is surrounded by crushed gravel, granite. You could easily host 3 to 5 'patio' trees in a natural configuration. Plenty of the most wonderful dappled shade imaginable.
You have a couple of flowering large shrubs already. Do you become anaphylactic if stung by a bee? I hope to quell you notions of 'no bees'...BUT there is another solution or two.
An awning or a patio umbrella is instant, colorful and will last until you get your perfect tree.
A simple pergola/arbor planted with golden hops will give you deep shade and fast. They flower in the fall and I've never noticed hordes of bees. Within the last decade I managed to get stung at least 47 times within just a few minutes...47 stingers got pulled out of me of bald faced hornets. I did go into anaphylactic shock but I've been stung since and no problems. Bees are our friends. If you don't bother them they won't bother you.
Bald faced hornets (that is what the doctor called them I thought they were black and white faced) were tiny, fly size, golden but my goodness, their sting packs a punch! Especially when one stings you on the top of the head after the horse has disturbed the nest and is getting away...these hornets FOLLOW...even galloping away, one got on top of my head. Thought we were in the clear and BAM! just one feels like a sledge hammer hitting you over and over again. (The stingers have their own little muscle, a pump to continue to pump the sting after being separated from the hornet)...each squeeze of that muscle you actually have to duck as if you've been hit by a sledgehammer. Ugh.
Bees are not at all a big deal, unless you put your ball cap on over a live bee in your hair (working around bee hives I was stung this way only once), still not a big worry. Gardens aren't gardens without bees. All plants flower. If bees disappear, we humans disappear shortly afterwards.
I have a few epipens just in case, but like I said, I've been stung a few times since and no problems for me. I am also out in the wilderness on horses covering lots of territory. Some of these hornets eat dead meat. I happened to climb up a bank and intrude on their recent cache. Oh my. I'll always be alert to meat eating hornets but never worry about bees. They are the good guys. Even wasps, the European kind are good guys and not at all aggressive. Bees are not at all aggressive.
I love your backyard and (I admit I looked up Raki...grins, I drink saki, never tried Raki...) I get it! AAhhhh!
If the bee thing is a deal breaker, the architectural arbor/screens with vines or a patio umbrella will fix you up quite nicely!
I just added a picture of a back yard in the Pacific North West. They had a thick beautiful lawn but big 800 plus pound Elk just walking across their lawn coming to a stop causes the sod to buckle just like carpet. So we got rid of the lawn and replaced with 4" crushed 3/8 minus gravel. Added the pavers to make a transition as well as a place for table and chairs. The 'peninsula' blocks the view causing a bit of mystery as well as enlarging the space mentally. If you can't see it, your imagination makes it up. Lots of 'rooms' are way larger than one big 'room' in our minds.
I would find out what hornet is in that large shrub. Let us know. You might not want to install a lawn. How old is your child? I grew up in Japan where the parks were crushed gravel, crushed granite not lawns. Kids love gravel.
Let me know what you think. I am trying to find my pictures of groves, patio trees to help with your visualization. I have filed my pictures poorly and I have thousands.
Hope this helps, I'll be back with more pictures and suggestions. Would like to hear what you think at this point...
Adding pictures for imaging...this one is for the 'arbor' and vine for shade...
shade by architectural arbor
olive mini grove
patio tree size and feel
transition from formal patio to informal patio
patio trees for Greece I vote for Redbud and Katsura
Amelanchier is my all time favorite 4 season beauty tree
Amelanchier Service or Sarvisberry first to flower in early spring profusely
a simple 'grove' in a backyard
power of professional lighting, these are multi stemmed patio trees breath taking
service berry 'grove' in fall