I have input from three different situations that are similar to yours.
First, me. My father was married three times, with my mom being his second wife. He legally adopted my older brother (who is biologically my half sibling) because he was just a baby when my parents met. Prior to meeting my mother, though, he had four kids from his first marriage. So I had four half siblings from his first marriage. One, a brother, died young. The other three all live within an hour or so of me and they are all a minimum of 11 years older than I am. I have met my half brothers on a handful of occasions, and I didn't meet my half sister until our father died 10 years ago. My father was very insistent on keeping his "other kids" separate from us and us from them. I don't know his reasons but that's how he was. I think he felt guilt about re-marrying and having more children.
I always felt sad about this, knowing I had other siblings close by geographically but that they are virtual strangers. I think it was, in part, because I grew up basically as an only child. The brother I grew up with was nine years older and moved out when I was just nine. Technically my "half" brother but never anything besides brother to me. He and I were (and are) very close but I always longed for a "big" family. I was VERY sad when he moved out (he got married). I never had cousins that I was close to, or much extended family at all. It was just me and my mother from early on. My Grandmother lived across the country and I wasn't close with my Aunt or Uncles (father's siblings). I spent minimal time with my father. I also always REALLY wanted a sister. My half sister does not want to have a relationship with me though. She hasn't come out and said it but my attempts to contact her have proved fruitless, even though I am in contact with her son (my nephew) who is actually close to me in age.
My half brothers are "Facebook friends" and that is it. It has always left me with a sense of having missed out. I still kind of wish we were close, that our kids knew each other, that my kids had them in their lives as Uncles / Aunt and vice versa but oh well. I spent some time with one of my half brothers when I was a kid; it was just a weekend but he was really loving and kind towards me and I always remember that weekend. He is the most "friendly" to me now but it's still just limited to saying Happy Birthday on Facebook or whatever. The other one is too much like my father.. ugh. I don't think there's a future for a relationship there.
Next, my wife; she has a very similar situation to yours, in that her father has, that she knows of, something like 15 kids with at least 13 different women. He even gave two of his daughters the same name, years apart! Creepy! And my wife also grew up as an only child, even though her mother also had kids -- six -- who were raised by their father. So, all total my wife has something like 21 (!!) half siblings and she grew up an only child. She would really like to know more about them but hasn't even managed to track them down or sort them all out. She did meet one half sister and an Aunt, but it kind of "fizzled" after the initial excitement. I think she would definitely go to a "reunion"/meetup if she could, but only if it was easily affordable. I know she wouldn't want to do it if it was a financial strain. There is still a strong desire to eventually know her half siblings though. And again, maybe it is because, like me, she grew up with just her mother, mainly, and she didn't have the greatest childhood. Maybe for us both it is about some fantasy about big, happy families. Who knows.
Now, my oldest son has half siblings as well, that he has never met. His father was one who "got around", and there are four half siblings that he claims but I suspect there are more around town that he denies! I joke with my son that he should make sure any girl he ends up with knows who her father is or she could end up being a half sister!
Anyway my son is 21, but when he was 18 his father (after 15 years of totally ignoring his existence) let me know that he'd like him to meet his younger half siblings (all younger -- they live with him and he is married to their mother) and wanted me to pass on the contact info to my son. I was kind of surprised but my son has zero interest in meeting them. The way he put it, it would just be
weird. He said "I have a brother already, it's hard to think of someone else as my sibling.." referring to his half brother that he grew up with (my younger son). So for him, there's just no desire, but I think (I hope he agrees!) that he has had a pretty emotionally fulfilling upbringing and a good relationship with his brother, and maybe that's it. I don't know.
As others have said, it is obviously a very individual decision and you shouldn't feel bad one way or the other. I don't expect or want my half siblings to feel bad because we don't have a relationship... I mean, it makes me sad but I am not blaming them for it. I think they possibly picked up negative attitudes from their own mother about their father's "new family", meaning me and my mom and brother, and that probably has an influence now that they aren't even aware of.
Finally; yes, it is sad when people just procreate willy nilly without regard to how this affects children all in different homes.
"Oh what a tangled web we weave...when first we practice to CONCEIVE!"