The post of our driveway gate protrudes about 10 cm onto the neighboring property

Yes, just one step at a time. That is already stated above.

And how is he supposed to judge that as a layperson?

As a layperson, I would measure the distance of the boundary stones to the neighboring boundary stones. That should at least serve as an indication.

Otherwise, the architect should first more precisely name the boundary where he thinks it is. In particular, he must clarify his strange drawing:

As proof, the architect hands us a drawing approved by the city of Cologne (with stamp as a copy) that is supposed to demonstrate that the post and the dowels protrude into the neighboring property.

However, the posts and dowels in question are not indicated on the site plan drawing.

Furthermore, I would rummage through my own documents concerning the land and construction and check for plausibility. For example, if the property is 15m wide, but the building on it measures more than 15m, that provides the next indication.

If the neighbor/architect bases their opinion on that, you won’t avoid a boundary determination.

If the client does not want an amicable agreement, then yes, you will have to do that. How much does such a determination cost?

And how is he supposed to judge that as a layman?
In the end, he thinks he is right, makes a fuss, and the neighbor is indeed right. Or vice versa, there is a stone on his side, he believes the neighbor is right but in reality he even has 10 cm more. Then he gives away land. In my opinion, a specialist has to be involved in any case.

Find the boundary points and measure them yourself once with a string if possible, just as describes.
If I then see myself as being right in the current boundary situation, I would politely and clearly communicate this to the architect and also show it and request urgent attention.
Then it would be HIS turn and he would have to prove the opposite to me by official surveying or similar. Whether he would bear these costs remains to be seen, but initially, I am not expected to doubt a set boundary point. That is precisely why it was set.
Of course, you are right, making a fuss is generally out of place, otherwise the OP would have an ulcer even before moving in.
If the architect could officially prove the opposite to me, I could also understand that the new neighbor wants to use their entire property. I fear that otherwise one would have to go to court and not really get justice there either.
But then I would no longer want to live there... therefore I would always look for the peaceful solution, wall or no wall.

I fear that otherwise one would have to go to court and wouldn't really get justice there either.

There would still be the question of whether a dismantling is proportionate. That certainly also depends on the overall situation. The court would have to assess that. But I like the idea of putting the ball back in the "other side's" court. Let them prove that the boundary was actually built over. What does the architect want to do, give the construction workers instructions to tear down posts and carport? Hopefully it should be clear to him that he cannot just do that. So politely inform them that everything stands on your own property and you would only accept it if a surveyor determines otherwise... By the way, with my property it was also the case that during the re-survey it was found that the neighbor had built over the boundary with his shed. Result: the property was divided, and he had to buy an 11m² piece. A dismantling was not demanded. (This happened when the city wanted to sell my current property as a building plot.) I would actually have liked to have those 11m², after all, the total is only 308m²...

Of course, one should not forget the neighborhood law here! If the original poster says no wall, then there is no wall for the time being, and I think there is much room for negotiation, and as already mentioned, the unlawful removal of the old fence also offers a lot of room for negotiations.

Has the house already been surveyed or is the appointment still pending? If the surveyors are there, they can also immediately say exactly where the boundary is. That is not a big deal for them and is more or less part of the scope of surveying the house anyway. It should definitely work out without much fuss.

Even if the method is not nice, it is still the neighbor's right to build something on their property. 10 cm is not exactly little, and over several meters in length, several square meters add up, which depending on the land price can easily exceed 1000€. Normally, the 10 cm probably doesn't matter, but maybe the space is also needed? There are also narrow properties and if you have to maintain 3 m on both sides, 10 cm are suddenly quite inconvenient.

First of all, it needs to be clarified whether the claims are even lawful. The burden of proof clearly lies with the neighbor.