but it also rather allows heat input through the sun from the outside to the inside.
And you really want that? Do you have any idea how warm windows get when the sun shines on them? Like radiators! Do you want to have radiators operating in summer? No? Then install triple glazing. It is better in summer and winter. In winter, when you need it, there is usually no sun due to the weather, or because of the shallow angle only from 11 am to 2 pm. The heat input is taken into account in the energy calculation, don’t worry, but it is simply a joke compared to the overall balance.
The argument about a poor insulation standard of the rest of the house is also questionable. The windows are usually the weak point. Even old walls and ceilings often still have better U-values than new triple-glazed windows.