I am confident that I don't need to worry about my grown up daughters and I'd take exception to any suggestion that I don't care just because I don't worry. I cared for them and educated them and watched them grow up into confident, practical, sensible adults. I still have one at home who has to grow through the teenage years, but I'm not worried about her either and hope I shall never have to. The prognosis is good at the moment (she's intelligent and sensible most of the time) and I'm keeping my fingers crossed carrying on educating her and hoping she'll turn out OK — be able to run her own life.
Most of the time I don't know the details of what my grown up daughters are doing and I don't feel the need to either. They tell me what they want to tell me when they want and I tell them what I want to tell them when I want. I'm never waiting for a call or a text or an email and neither are they. They are like other adult friends — except with the qualifier 'special' inserted before 'friend' — and as with my other friends, I would hate them to take me for granted just as much as I'm sure they'd hate me to take them for granted. I feel sorry for people who don't have this freedom in their relationship with their grown up kids. It must greatly add to life's stresses not to have it.
To put it another way, my view is that my kids don't owe me anything and all I owe them is what I can give them without mucking my own life up. Constant worry about them would muck my life up. How would I feel if they'd been disasters instead of successes? Who knows?
I'm beginning to wonder if this isn't really about parents' confidence in themselves as effective parents as well as their confidence in their children's ability to lead successful lives as independent adults, by which I mean without mum or dad having to step in and help out except in emergencies or drastic life shocks such as happened to GA's daughter. By all means we should continue to relate to our children as much as we and they want and is convenient, but I certainly would be horrified to think that my adult kids needed my help just for getting on with their ordinary lives. That would mean that I had been a failure as a parent. Nightmare thought! All that hard work gone to waste! 
Here's an example of what I would call unnecessary worry. I was climbing in North Wales one time with a guy who was twenty-five years old and, at the time, a doctoral student at university. We had made our way part way up a rock face only to find that we couldn't go any further so we back-tracked. He went down first. I waited on a four-inch ledge. I saw him about to put his weight on a pillar of rock that I had noticed was wobbly on the way up, but before I could shout to him not to put his weight on it, he had, and it had toppled, he had fallen, and part of the rock had landed on his ankle and broken it. The rock fall also removed my easy way down. Anyway, to cut the story short, we got help (shouting; no mobiles), he was helicoptered to Bangor hospital and I was helicoptered to where we'd left the car, from whence I drove to Bangor. The ankle needed an op for a pin to be inserted so I decided to ring his parents just to let them know. His mum came haring over to Bangor from Birmingham!!!!! WHAT?! For a broken ankle?!!!!! Ye gods! To be honest, I don't suppose I would have even told my parents until well after the event.
He told her to go home. We continued our holiday with him on crutches. We visited a lot of cafes
. My friend emigrated to Canada as soon as he could after completing his doctorate and I think he still has the pin in his ankle.