Advice needed re gutter repair

[Barbs In Trouble]

The gutter on my extension was a lead lined box type (I think). This gutter takes a fair bit of rain water as a down-pipe channels water from the main house roof into it.
A fault developed in the gutter: the lead split and water was pooling at the joint with the main house so I got some builders in to repair it. However, I have grave doubts as to the state of the repair.
To correct the pooling issue, the builders laid a board at at a slight angle lengthways to facilitate run off. This was then sealed with some burn-on felt of some sort.
The issue I have now is that all the water from the downpipe now runs up under the tiles. It may appear further down the gutter; it may run down the house/extension wall. I don't know. The board they fitted is angled inwards towards the tiles so the natural flow of the water is under the tiles instead of in the gutter gully (the gully doesn't actually exist anymore due to the angle of the board).
I've challenged the builder but he insists water cannot enter through the roof as "there is a complete seal outside tiles and beneath tiles". I don't understand and am having difficulty getting a proper explanation. I guess he's burnt the felt onto the existing felt and laid the tiles on top so I assume the water tightness depends on how high up under the tiles the burn-in felt goes. However, with v heavy rain, I still worry that water will go up under the tiles +/- some capilliary action.
I'm not a builder and I'm too old to be climbing up ladders. Should I just accept the builders word or should I be seeking some form of redress? Your advice will be most welcome.
It was raining when I took the attached photo. (V scarily hanging out of a window).

 

[Nige F]

A sure fire way would be to take the rainwater pipe down onto the wall and along - unfortunately out + back round that vent pipe - along to the end of the building and into the drain. Then the volume of water on the roof "gutter" will be greatly reduced.

 

[Barbs In Trouble]

Thanks so much for replying.
My concern is that the gutter they've fitted is not fit for purpose. I paid a lot of money for this. At my age, I don't really feel safe climbing ladders etc hence why I employed a builder.