I connected it directly to the PC via USB<>TTL serial adapter and with own software, u-center, it works fine - locks satellites, displays position, displays time.
As I understand, I need to care for this line only:

$GNZDA,202437.000,21,07,2024,,*48

So I believe that issues are related with maybe voltage levels, I'll try to add some resistor dividers tomorrow and will check again.