The short form of my long post would be: "You can get drainage by either raising the bed above grade (a truly RAISED bed) or by lowering the temporary water table by providing rapid drainage through a trench, so that excess rainfall escapes before it water-logs the amended soil.
I don't think it takes hours for root hairs to drown when underwater (depending on the plants). Remember that after every drowning event (rainfall) the plant has to get rid of rotted dead root tissue and then rebuild it before the plant can take up as much water and minerals as it used to. All its energy goes into re-building the root system after every rain.
The photo looks like you don't have much slope to work with. Maybe forget trenching. Maybe make the new bed partly raised like the area behind and to the right of the hammock? Like an extension of that bed? Maybe only 12-18 inches tall?
One thing that I do when I want more drainage but less digging is to put a partially sunken walkway around a partially raised bed. Loosen the soil or double-dig the bed before digging the sunken walkway. Throw soil from the walkway up onto the just-turned bed, making it deeper at the same time you dig the footing lower.
I do trench to drain the walkway, but if you have no "low spot" to trench TO, your sunken walkway could form a moat after each rain. The bed would drain rapidly into the moat while the roots in the raised bed stayed alive. Then the moat can take an hour to drain if it wants to.
If your clay soil drains slowly, digging a hole below grade (lower than the surface of the clay) will cause that hole in the ground to fill with water in a heavy rain. Clay "perks" very slowly, and rain can fall pretty fast.
If you have a sunken bed in a below-grade hole (amended soil in a hole below grade), it will turn to mud in the first heavy rain, and that will probably collapse the amended soil structure as well as drowning roots.
If the clay drains faster, and you don't have heavy rainfalls, you might be OK with a bed that goes below grade. You'll soon find out. It all depends on "how heavy" your clay is, how fast your rain is, and how tolerant your plants are of having drowned roots.
If the seed packet says "needs well-drained soil", don't plant it in an un-drained sunken bed, unless your soil is at least as open as loam or sandy clay loam. Clay, clay loam and sandy clay are not "well-drained soil".
Unfortunately, not even sandy gravel is "well-drained" if it is the back fill for an un-drained sunken bed surrounded by poorly-draining soil. You would just have a puddle with a nice sandy gravel floor.
>> And my clay is not THAT bad that water sits on top of it for hours on end.
The grass is clearly not drowning. Maybe your soil does drain fast enough for a sunken bed to have a chance.
However, the rain water MIGHT be running off in a thin surface sheet you can't see, under the grass, down a very slight slope. If that is going on, squint hard and figure out what the slope is. BLOCK any uphill runoff from entering any sunken bed.
Digging a hole would tell you as soon as you had a heavy rain. Sue has the right idea. Real estate agents call that a "perk test". I guess in areas other than the coastal PNW you might have to fill the hole from a garden hose. Where I live, just wait a day or two.
I'm a big fan of sufficient drainage. It doesn't help to put fast-draining new soil into a puddle.
The BED has to drain well, not just the soil IN the bed.
