Does the ventilation system really pump humid outside air into the house?

But you should anyway only run your ventilation system at the minimal level during the day in summer, especially in high summer. Because besides the humidity, you also bring warm air into the house. The heat exchanger can hardly prevent this.

Wrong, modern ventilation systems recover up to 90% of the heat. This works at -10 degrees outside and 21 inside. But it also works the same at 30 degrees outside and 21 inside. It is even noticeable that the delta T in summer is much smaller. I have already calculated several times here in the forum how small the heat input through a running ventilation in summer is. So let the ventilation run both in summer and in winter to maintain a pleasant indoor climate. A control depending on humidity and CO2 can make sense. Temperature should not cause any changes due to heat recovery.

I have already calculated several times here in the forum how low the heat input from running ventilation in summer is.

Links would be great...

Wrong, modern ventilation systems recover up to 90% of the heat.

But what do you do if it's 26 degrees inside and 30 degrees outside? Then the ventilation sends you humid air (assumption: there was a thunderstorm or heavy rain 30 minutes ago) at 28 degrees (a mixture of 26 and 30) inside, which still leads to an increase in temperature and humidity in the house. So running the controlled residential ventilation is not advisable, is it?

Links would be great...

For example here: Feel free to ask if you have any questions about it.

But what do you do if it’s 26 degrees inside and 30 degrees outside?

All the better. Only a delta T of 4 degrees. With active heat recovery, therefore, a maximum of 1 degree. Thus, there is no (or hardly any) heat gain.

Then the ventilation sends moist air (assumption: there was a thunderstorm or heavy rain 30 minutes ago) with 28 degrees (a mix of 26 and 30) inside, which still leads to a temperature increase and higher humidity in the house. So running the controlled residential ventilation is not advisable, right?

Here you are now a bit imprecise and mixing temperatures and humidity. I had written that it can indeed make sense to adjust the regulation when there is humidity, as long as CO2 remains low enough. However, this has nothing to do with temperature (apart from the influence on relative humidity).

But what do you do when it’s 26 degrees inside and 30 degrees outside? Then the ventilation sends you moist air (assumption: there was a thunderstorm or downpour 30 minutes ago) at 28 degrees (a mix of 26 and 30) inside, which still leads to an increase in temperature and humidity in the house. So running the controlled residential ventilation is not advisable, right?

You don’t do anything, because practically nothing happens. You already had the moist air outside and inside before.
Turning off the ventilation system for half an hour is not worth it.

And you can safely neglect the temperature increase caused by controlled residential ventilation. During a month of constant outdoor temperatures above 30 degrees, I never got it above 24.5 degrees inside (somewhere the sun still came through the windows, I don’t like completely darkened rooms).
26 degrees inside requires several weeks of Sahara summer with nights still at 25-30 degrees.

And even if some moisture gets in - that is still less with a ventilation system than with chaotic ventilation through windows.

I have a differential humidity control built by myself in the garage. Maybe someday my ventilation control will also get a bit more intelligence in this regard - it’s nice when you have the wiring in the house and just need to connect a more modern control unit.

26 degrees inside definitely requires several weeks of Sahara summer with nights still at 25-30 degrees.

In a well-insulated house, yes. Otherwise, easily up to 28/29 degrees (house built in 2012, minimal standard at the time).