Why do we require social interaction/engagement?
I think you phrased this well, but you are correct that it is a hard question which could be answered in many different ways which, perhaps, may not satisfy you. It is a "why" question so let me give some speculative answers:

-an evolutionary psychologist might say that we have evolved to seek social interactions because humans in groups are more successful (e.g., they can mate, raise offspring together, defend each other from predators etc). It is hard to refute this kind of explanation, and indeed many/most species show a desire to interact, although some niches may have encouraged more solitary behaviour.

-a behavioural neuroscientist might say that we seek social interactions because they are rewarding on a biological level, for example because they release oxytocin which has a range of positive effects on physiology.

-a social or developmental psychologist might focus more on the way that social interactions change self-reported well-being, or on the qualities of the interaction which are most important (perhaps in line with your bolded question). While it is true that particular types of (positive) interaction are best, even minimal contact increases a feeling of belonging. The developmental literature on social enrichment might be a good place to look if you want to find out how to make satisfying interactions.

So, like most "why" questions, it depends what level of analysis you are interested in, but all of these are probably true to some extent.