I'm not sure that I entirely agree with the premise of the question.
Firstly, I think there is a significant element which is probably innate/genetic. So I'm not sure if confidence is really something that is entirely within a parent's gift. Though I do think that parenting can help.
Secondly, I'm not sure that I agree whether it's a question of nurturing them and making them feel secure versus pushing them out of their comfort zone. I think you have to do both. Pushing a child to do something that makes them feel uncomfortable without any support is just cruel. Giving them the scaffolding and support that can help them to take that leap of faith is empowering.
My dd has always been extremely confident - I envy her for it to be honest! She doesn't have a hint of arrogance about her, but she does has very high self esteem - she genuinely likes herself as a person and she assumes that others will like her too unless they tell her otherwise (which almost never happens!) . She is very accepting of her strengths and weaknesses and readily acknowledges areas in which she needs to improve, while being equally accepting of others' strengths and weaknesses - fundamentally, she seems to have grasped the fact that nobody is perfect and that it's OK not to be perfect. She is willing to give anything a go and throws herself into any new situation with enthusiasm, she doesn't hesitate to put herself forward for stuff and she doesn't tend to be afraid of looking stupid or showing any vulnerability. I think this willingness to take social "risks" has enabled her to create lots of opportunities for herself that have merely helped to strengthen her sense of confidence and self belief.
I do think a lot of dd's confidence is related to innate personality traits. However, I think some of her early experiences probably helped her a lot as well. She was very much wanted and loved; dh and I were happy and in a stable during her early years, and we loved being parents. She got a lot of time and attention and a lot of physical affection. We listened to her for hours and always valued her opinion. We were also lucky enough to have a very loving and supportive family and to be living within an amazingly connected and supportive community - she had a lot of people around her in her early years who were interested and affectionate, and I'm sure that this shaped her outlook on the world and her expectations about how others would respond to her.
DD has told me that I always readily admitted when I got things wrong and apologised, and that I emphasised often that nobody was perfect. She thinks that this was important in helping her to accept her own vulnerabilities and flaws. She also suggests that the way any problems were handled was helpful - rather than rushing in to solve problems for her, we talked a lot about how she could solve them for herself, rehearsed what she might say to people and how they might respond etc.
DD was lucky enough to be naturally very academic so she experienced a lot of early successes which probably helped to build her confidence a lot - she felt competent and capable. I never pushed her though, and always emphasised that I valued effort over outcomes. I was very keen that she should learn how to fail well (having feared failure myself through most of my life) so I actively sought out opportunities for her to do stuff that didn't come too easily to her. Stuff that pushed her out of her comfort zone and stuff that she needed to persevere with to get better at it. I think this really helped her resilience but it wasn't ever a question of just leaving her to sink or swim...I wanted her to know that she was actively supported effectively to do the things that she found difficult. That might mean helping her practise for hours or paying for 1:1 tuition etc. And she was always in control of the process so she knew that she could stop if she had had enough, but we had lots of discussions about how good it would feel when she mastered X, how important it was to persevere and work for stuff, how proud I was of her resilience and determination etc. It took her 10 whole years to learn how to do a cartwheel, but she got there in the end, and I was so bloody proud of her sticking at it...way more proud, tbh, than of her string of perfect exam results which came to her much more easily.
DD learned quite early on that taking those small risks - like putting yourself out there to audition for something even though it felt really awkward - might be a bit uncomfortable in the short term but led to really positive outcomes - like getting a really good part in the play. We talked a lot about how most people are too worried about themselves to notice how nervous you are. She also learned that being "all in" with activities and showing real commitment typically yielded good results, and I think this helped to build a strong sense of self efficacy and self belief. She learned that if she worked at stuff - even stuff that didn't come to her easily - she could make it happen. That is incredibly empowering.
We also talked a lot in those early years about friendships and family relationships etc, how to be a good friend, how to handle tricky issues etc. Again, I think dd learned from an early age that, if you invest a lot in relationships with others, you typically get a lot back. This really helped to build her social confidence. She learned that, if you make a real effort with people, they will generally like you.
In the end, I think it becomes a bit of a virtuous circle. Every positive interaction, every nice comment, every little achievement, every challenge overcome... they all help to affirm that you are a likeable, capable person and that it's worth making an effort. That gives you the courage to talk to new people and attempt new (perhaps bigger) challenges, and it also gives you the determination to keep trying at the stuff which you find difficult. Often, you succeed again and so your confidence and resilience grows all the more.
Of course, there will be knocks along the way. Failures, setbacks, disappointments. But if you have a secure sense of self esteem and and a general sense that the world will usually respond positively to your efforts, you will typically frame these as minor obstacles to be overcome rather major issues that will get in your way. I've noticed that my dd will typically find the positives in any situation, even when things haven't worked out as intended - she will acknowledge any disappointments and allow herself to feel them, but she doesn't dwell on them. She tends to notice the unexpected benefits and opportunities instead of focusing on the stuff that hasn't worked out as intended. Looking back, I think this is something that I used to do with her when she was little as well, and it seems that she has now completely internalised it!