By insecurity I just mean it in the everyday sense of someone worrying a lot about how other people feel towards them and being afraid of being rejected, excluded, ostracized etc..

That's entirely social insecurity. One could also be insecure about one's level of competence, regardless of the opinions of others.

My thinking on this is that if someone is not afraid to feel disappointment or loss they won't be insecure.

I think that's some of it. Failing to face a possible outcome and accept it makes anticipation worse. To face it is to usually see that you will survive, and it won't be so bad.

However, some insecurity is an ingrained emotional response to actual events. If one grows up in a hostile or treacherous environment, your emotional reaction will likely be appropriate to that threat, with little regard for a change to a safer and less treacherous environment. Also, one thing I've recently considered, is that the habitual act of looking for threats makes the world seem more threatening through the availability heuristic, making you feel less secure than would be appropriate. Notice that heightened vigilance for threat is a natural response to a threatening environment, but it would leave one feeling less secure even after the threat is removed.

In both failure modes, there is a cost to anticipating future failure, whether in the pain of that anticipation, or in estimating the world to be more threatening than it is, again leading to painful anticipation, but also inaccurate predictions of risk that create suboptimal action.