In my estimation no. No these types of "friends" are not worth keeping around but then again i'm just a nihilistic cynic so my worldview is quite different to a majority of folks.
This situation has described my life to a tee. Ever since I was a kid, if I wanted interactions with childhood friends, I would have to initiate it. If I want to talk with co-workers now, I have to initiate the interaction. Socialization always just seemed to happen naturally for everyone else but for me? I've had to work for it my entire life.
I'm currently on LOA from work, I have been gone for 31 days so far. I left to manage my depression and other issues. A few of my co-workers whom have my phone number and were initially so adamant about being my friend; have not called me once or texted me once to ask how I am doing. I guess I pushed them away. I've told them that we are not friends but co-workers and gave them my number for work related inquires but still, no interactions, even from the "kind" ones.
I'm not worth conversing with. I have made myself unapproachable in many ways and I'm not really sad about it. I mean, I don't deliberately try to be crass or mean in any way but once I start talking to most people, I never get that sense of fulfillment; I don't know how to describe entirely what I am looking for in other people in order to make friends with them but I do know that I haven't found it yet. To be honest, as arrogant as this makes me seem, most people also don't really have anything interesting to say so I have made peace with my limitations and often reside with my own thoughts rather than talk to people.
I don't have any friends in real life or online but if I did and I was still in the predicament where I always had to start the interactions, I personally would just stop them altogether and once again reside with myself.
As it pertains to anyone that actually wants friends, I guess that is their lot in life and they just have to deal with the fact that they always have to start it first.
"We live in a kingdom of bull****. A kingdom you've lived in for far too long.." -Mr. Robot