Circumvent 433MHz antenna printed on PCB
Once done, there's no way back anymore, so maybe there's a less
intrusive solution

I wouldn't hesitate to buy the evaluation board (see data sheet) and test this out first: -

Then you are not risking unnecessary damage to your main circuit. I read the data sheet and there are parameters unlisted such as the chip input impedance. Also, you probably cannot determine the pi-network components on your original board to infer what the printed antenna impedance is. If the printed antenna did have a clear-cut impedance value then things would be a tad simpler of course but, as it stands, it's guesswork.

Is this antenna a valid choice?

It's hard to tell because the printed antenna impedance is what the pi-network is matching the chip with and any external antenna will likely not be matched to that pi-network the same as the printed antenna. However, the one you linked is not suitable because (apparently) it has a signal amplifier built in. Choose an antenna (a monopole, quarter wave) that has a recognized data sheet and is bought from a reputable supplier for this type of product (not Amazon).

Could I solder the external antenna onto the first solder point and
leave the printed antenna intact?

You could do and, due to the simplicity of such an action it's worth a go but, results may not be that clear because you are significantly affecting the pi-filter input impedance by putting two antennas in parallel.

As for GND of the external antenna: Can it be soldered onto any solder
point carrying GND on the PCB or does it have to be the GND pin of the
SYN531R

I'd buy the evaluation board and split the main board at the data output point.