I agree with you that at the individual level, forgetting things is important for sanity and wellbeing.
When it comes to the culture of a country though, I don't think we should only preserve what we currently consider interesting. The purpose of an archive is to preserve things which may become interesting in the future.
Consider a society of philosophers, poets and artists who create amazing pieces of art. Then, consider a change in that society where the populace is more interested in horse races instead. Now all their writings become about horse races, the biographies of those horse racers, etc. They dismiss the old stuff as uninteresting. Should they destroy the books written by those philosophers and instead recycle the paper for printing stories about racing horses onto it? For us in the future, such a destruction would be quite sad. It would be silly to assume that we don't have present biases either. Maybe back then, everyone was familiar with the poets so there was no need to preserve their art as everyone already knew it. And why teach it to children when they are more interested in horse races?
Many archaeologic sites have been visited by pillagers again and again over the centuries (continues until today [1]). What are they taking? The interesting stuff they can sell. Too often, only the "boring" stuff remains to contemporary archaeologists. How happy they are when they find a receipt written into a clay tablet!
For decades, no piece of Bach was performed publicly. Now he's considered as one of the greatest musicians of all times. People could have tossed away his manuscripts as uninteresting yet they didn't. Another example is how many scenes of the Metropolis movie were cut during distribution and the original version only survived thanks to an Argentinian museum.
It gets even worse when political or religious arguments come into play. Greek art was considered heathen lore and only survived thanks to muslims preserving it. Mayan books were burned by missionaries. IS has been bombing many ancient sites it had control over. All these folks consider the stuff they destroy to be harmful and uninteresting. After all, a good muslim is only interested in qran, a good christian only in the bible. Fortunately we at least preserve the writings of the past centuries that we consider politically incorrect today instead of destroying them, but there are stories of british archives even in the present day "misplacing" evidence of crimes committed during colonization.
[1]: https://www.abc.net.au/news/2018-10-21/egypts-3-billion-doll...