the slope away from the terrace and drainage under the slabs is sufficient
In heavy rain events, no.
Do you have a barrier-free/close-to-the-floor lift-and-slide door? Then take a look at the fine print, driving rain and the like are excluded; those things are never 100% tight. I feel considerably more comfortable with a channel, especially since the terrace faces the weather side.
With a roof over it, you could rather speculate about doing without it.
So it costs a lot of money, you can easily calculate 500 euros per door. For us, it would have been 2500 euros. Doing it yourself costs you only about 300 euros for materials. So over 2000 saved! Labor effort about 6 hours.
A bucket of sealing paste costs 50€, the work is a joke. What should cost 2500€ there?