It is not necessary it’s just the natural line but what I would suggest is to have some drip trays (felt) Installed it will help prevent the rain coming through the gap.

I think these have been fitted incorrectly as there shouldn’t be a gap, as like you said, water will drop between the gap. The roof felt should overhang and sit in the gutter bridging the gap and this seems to have been too short. I would suggest to have eave protectors fitted. These slide under the bottom row of roof tiles and sit within the guttering to allow the water to run into the gutter.

Hard to tell without seeing properly but daylight probably is because the felting has rotted back and nothing much between tile and gutter, you can quite often fit an eaves tray to stop this

This new guttering is useless, as seen in your video, rainwater is not getting into the gutter and falling through the gaps!
You need to install “eaves trays” all along the bottom of the roofline (or where there’s guttering) these are basically black plastic sheets approx 1m in length x 300mm wide, they slide under the last or bottom roof tile and sit nicely into the gutter. You have to overlap them by about150mm to form a continuous Chanel.
They cost around £5 each.
Looking at your house, I would consider hiring a tower scaffolding to carry out this simple bit fiddly job. Best of luck

Usually if you see daylight behind gutters to roof may be felt could be rotted you can easily fix with drip trays