Hi there Calloy and a BIG welcome to our forum.
Thank you for a beautifully written post - that must have taken some courage.
I see that Sophie our community moderator has suggested that you may find one of the Beyond Blue support services helpful.
I am one of four brothers, and have had a similar situation with my only sister (two years younger). I thought I would share how things went with you.
All my brothers used to pay out on her and I in particular said something hurtful one day that I later regretted. So we didn't have any contact for well over a year.
Anyway one day when in New Zealand I called her and said I would like to catch up and discuss care for our aged father. When we met I confessed I actually wanted to talk about our relationship as I valued and respected her as a person and wanted us to chat more. I was quite ready to apologise for what I said but she brought it up first, said there was no need to apologise and how much she appreciated me making contact again. Our relationship has been great ever since and I am so glad I made that effort.
In the case of your sister it may well be that her own life is full with looking after the teenagers, she could even have issues that she hasn't raised with you. You have shown courage to write this post - I suggest using that courage to contact her again. Playing the 'Its your turn to contact me' game rarely works. Something that may help you is preparing to ring her first by making a list of things you want to cover. This could help overcome any anxious moments. The first could be about how you love having her as a sister and notice you haven't been in touch much lately and wanted to chat more often. The second could be something like you were just thinking about something the two of you got up to when younger, how much fun it was, and thought you'd ring to have a laugh about it. And how much you miss those days together with her! Do you get where I am coming from?
Keep it positive, never mention her not contacting you as she will most likely raise it herself. Be sure to ask how the kids are and how their holiday went.
So yes, don't let ice form over you relationship and give her a call. Be very positive, tell her you love having her as a sister and express interest in her family. Bring up some positive experiences you have shared! And that she and her family members are welcome to stay with you anytime.
I sincerely hope this helps - I'd love to hear how you get on!
Bye for now, The Bro