Buy the land first and then finance it?

The most important question is about a building obligation. This is actually always the case for plots owned by municipalities. If not, they have problems getting rid of the plot and it is too expensive. If it is private, why wasn’t it sold during the 1% interest rate times. There used to be a guideline of 100-110 times the monthly income as a loan. That would mean a limit of 400,000.-- debt for you. But the myth that you can’t live in 20-30 year old houses and have to renovate at least €200,000 can probably never be eradicated.

no building obligation, since private sale

Is it building land? Why hasn't it been sold for a long time?

I believe everything that needed to be said has already been said, just not by everyone yet.

Basically, the question is how important owning a home is to you. Are you part of the 'work hard, build a house' faction, or more like 'I live – and consume – today'? Apparently, you can save. If you’re willing to forego vacations, parental leave, (new) cars, restaurants, etc. in times of need, quite a lot is possible.

Even with >8k net per month, we don't want a monthly payment higher than 2k. Because we go on vacation two to three times a year, spend a lot of time with the kids on weekends, like to indulge ourselves occasionally, and ultimately don't want to 'enslave' ourselves with a higher monthly payment for 10 or 20 years. But everyone’s mindset is different.

We bought a plot of land from our own funds because it happened that way and it’s in our absolute preferred location. Our goal is to build soon. But if we had to wait another two years, that would also be okay. We could also sell it anytime without major losses. And we would still be happy in a rental house ...

A plot of land is quickly bought,

...and quickly sold again. Provided it is building land.

Is it building land? Why hasn't it been sold for a long time?

That question arises! There are also plots that are only garden land or land with expectation of development. Stay away!
…

Net household income from September is expected to be €3,800 … Until recently, we mostly placed our monthly savings in a daily allowance account (due to the desire to buy a house), the other part was invested in ETFs, stocks, and gold

That is hardly noticeable between the lines. €3,800 is definitely too little. And then additional saving on top of that.
Of course, there are dual incomes that make it possible to invest both in property and additionally in securities at the same time. I don't see that here at all. It is not bad either. Most only rely on one because they cannot do both.
Reflect and decide in which direction you want to go. The fact is: anyone who wants to build also needs a plot. That would be the first step. Additional saving does not work with this income.
But:

Our family is putting a lot of pressure on us and says that we absolutely should buy this plot.

Don’t let family or friends pressure you. Let them buy and finance you… that would be the next answer if they get annoying.

If it's from a private seller, why wasn't it sold during the 1% interest rate times.

For many people, it was unimaginable that interest rates would drop again.

But the myth that you can't live in 20-30 year old houses and have to renovate at least €200k will probably never be eradicated.

You brought that myth into the thread. There are existing houses that were properly built and maintained. However, these are rare on the open market. Realistically, you usually get a 50-year-old house where the original bathroom is still falling apart. Not to mention the renovation obligations that will come in the next few years. These must definitely be factored in for existing properties, and with an energy-efficient renovation, you easily burn through €100-150k.

We also do not want more than a 2k monthly rate even with >8k net per month. Because we go on vacation 2 to 3 times a year, spend a lot of time with the kids on weekends, like to treat ourselves sometimes, and ultimately do not want to 'enslave' ourselves with a higher monthly rate over 10 or 20 years. But everyone ticks differently.

Being UNABLE to pay a higher rate is quite different from not WANTING to. If you feel like it, you can repay or save up a lot in a short time. For the OP, however, it is about survival when building and the rate really hurts or is barely affordable.