Would filling this part of my patio with drain rock improve drainage, at least in the short-term?

I moved into my house this time last year, so I don’t really know much in the way of its history, and how things have been done etc.

In the back garden there is a patio, and this patio is split into two levels. Along the part of the patio where the levels split, there is a section that is filled in with soil and small bits of gravel at the lower level:

It seems as if this lower part of the patio is sloped in the direction of this section, which makes sense as this moves water away from the house. However, water doesn’t seem to drain particularly quickly. When the rain is particularly bad we get a fair bit of standing water in this lower part of the patio. Here is an image of the water collecting, though it has been much worse than this in the past:

I’ve been looking into this a bit online, though I don’t really know anything about laying a patio or even drainage, but I am trying to work out if I can improve the situation before we get the patio completely re-done, as the rest of the house is much more pressing in terms of work that needs to be done (though having water issues like this close to the house is a concern, there are no damp issues with the house at the moment, and i have yet to see it so bad that the standing water has reached the back wall of our house).

From what I had read, I started to think that this part of the patio was a french drain, where the water was being redirected elsewhere (though I’m not sure where to exactly). So I started to dig up a small portion of the drain to see if this was the case, expecting to get to a drain pipe of some sort, however I dug 8" down, and have yet to reach any pipe. I may need to keep going, but I’m not so sure.

With regards to the manhole cover you can see in the image, I don’t believe this has anything to do with rainwater drainage, I can take the cover off and see there are no blockages, and the drain underneath is connected to our waste water output, I can see water coming through from the house when it is used, and there is only one place for water to come from under there. The drainage for rainwater from my guttering seems to be being routed somewhere else.

So I suppose I have a couple of questions, the first would be, is this a french drain of some sort? Or is it simply just a bit of the patio that is filled in with permeable materials in the form of soil/gravel, allowing water to into the ground under the patio? If not, could anyone explain what it might be? I appreciate none of you can actually dig up my patio to have a look, but I’m sure someone here would be able to hypothesise something.

My plan (having read about french drains) is to dig out all of this silty soil and gravel and replace it with larger stones, contained in landscaping fabric. I am thinking of digging out the whole thing down to about 8". Will this have any effect at all in terms of drainage? Even in the short term? Is this a bad idea/waste of time? Or would this potentially alleviate the problem, even if only slightly? I know this wouldnt be a French Drain without there being a drain for the water to go, but I guess there would be more space for standing water to go to while it dissipates, within the rocks, as opposed to this compacted silty soil.