I really feel you on this, being a FTM with all these decisions at once is genuinely overwhelming, and it makes complete sense that bedside manner and a calming presence matter so much to you, especially going into labour with some anxiety.
From what I've heard around, Dr Lee Jia Min at KKH has a really warm and gentle reputation, a friend of mine delivered under her and said she was patient, kind, and made her feel very at ease throughout, which sounds like exactly what you're looking for. Dr Manisha Mathur also comes up quite often when mums talk about wanting a thorough, attentive doctor, though she can be harder to get an appointment with, which suggests she's well-regarded. On the KKH private vs NUH question, some mums feel the private experience at KKH is quite different from subsidised, more continuity with your own doctor during labour, which for a FTM who wants that calming presence in the room could genuinely make a difference. Given your priorities around natural birth, delayed cord clamping, and skin-to-skin, I'd suggest shortlisting one from each hospital, then calling both clinics to ask directly how the doctor handles birth plans, that first interaction with the clinic staff can already tell you quite a bit about the practice culture.