Flat roof leaking, repair or build a roof structure over it?

Simply put a slanted roof on it? I don't imagine that architecturally so simple. Somehow it doesn't fit the house.

Does it have to stick cold because otherwise the insulation board would burn?

Yes, and it is cold self-adhesive, the foil is peeled off and it sticks immediately, when the top layer is welded on, the cold self-adhesive gets heated again and thus adheres firmly to the insulation.
The membrane is 1000x3 mm x 10 m/roll, TOP elastomer bitumen with glass fiber-polyester fleece insert, cold self-adhesive.
Secure seam connection through removable longitudinal edge strip.
Bitumen to bitumen bonding.
Direct installation on Thermazone EPS and Thermazone PIR aluminum possible.
Proven for over 30 years.
Price approx. 6-7 euros per sqm.
Then at least a PYE Photovoltaic 200S5 welding membrane on top (welded) and the roof is sealed. A small tip: welding membranes are not all the same, there are differences in composition, materials and thickness, e.g. thickness 5 mm and 5.2 mm carrier layer, bitumen etc. The price per sqm ranges from 7.00 to 18.00 euros for 1 sqm welding membrane.

440 euros per m2 is extortionate.

What should one pay 440 euros for? What all is included in material and labor? For 1 sqm?
Prices for materials:
VA primer 30 liters is enough for approx. 100 sqm, about 30-40 euros. Manufacturer differences in quality ;)
Vapor barrier welding membrane per sqm about 5-10 euros. Manufacturer differences in quality ;) thickness from 2mm to 4mm.
Flat roof insulation boards thickness/minimum 160mm, better 240mm price per sqm 25-45 euros. Manufacturer differences in quality ;)
Self-adhesive (welding membrane) roofing membrane that is glued onto the insulation 6-8 euros per sqm.
Top layer welding membrane minimum a PYE Photovoltaik 200S5, better options exist but they cost three times as much ;) price per sqm 7-18 euros. Manufacturer differences in quality ;)
Now calculate yourself what even the cheap material costs per sqm.
If you only buy good material... you're quickly spending twice as much money.
I sealed my entire roof 120 sqm myself in best quality including flashing, wood, gutter, etc. Only for material 10,000 euros. [ATTACH width="456px" type="full"]57405[/ATTACH]

What is one supposed to pay 440 euros for?

The house is located in the immediate vicinity of Munich, which probably plays a role. The carpenter could start the work at the earliest in August. Regardless of the carpenter, I have contacted the craftsman with the best reputation for flat roof renovation; the waiting time for execution is two years.

Just put a slanted roof on it? I don’t think that’s architecturally that simple. Somehow doesn’t fit the house.

The look also worries me a bit, but the decisive factor is what helps me sleep better. The rainwater would then drain outside, an advantage, because the downpipes already showed leaks at the opening cover in the basement. Admittedly, that can be fixed, but it remains a potential problem area. I prefer the rainwater to be outside. A conversation with a roofer is pending... I’m open to solutions...

Mindset. You have a flat roof, it worked and works even better nowadays. What rational reason leads to spending multiple times more to get rid of the flat roof?