I think all this has been covered but I just wanted to add my experience.
With my first child as a baby I found the HVs at best, pointless and harmless and, at worst, upsetting and incorrect - eg the ones that tried to force me to formula feed and told me my fully bf ds was "failing to thrive" because he was on the 2nd percentile. They wouldn't listen to me when I told them the graphs in the red book were based on WHO formula fed data rather than graphs for breast fed babies (as they were at that time 10y ago). He wasn't. He, and I, are just little and he is now a perfectly healthy 10y old, a head shorter than his peers. This was extremely damaging to a first time mother trying to bf in the face of much negativity. Everything I learnt about bf and looking after a baby I learnt from NCT classes and friends.
On the other hand, the one that came to the house after the birth was perfectly pleasant and helpful and gave me the red book which I think you really do need. The 2 and 3 year checks were also inoffensive and quite helpful. They cover things which I think you would really struggle to get a private paed to bother with - developmental milestones for a normal child etc.
With dc2 we saw them a handful of times, dc3 twice (get red book and 2y check). I really think this is your best course of action rather than officially opting out.
I do agree with you though, a healthy baby with 2 MC professional parents probably has little need for HVs and with anything medical, I went straight to my GP and referred to paed if necessary. My dc2 had a heart murmur and has hypermobility and dc3 had silent reflux and now has lactose intolerance (both of which the HVs spectacularly failed to pick up on). I got referred to a paed through my GP.
However, I am not sure what kind of access you have to a private paed, but after the birth, in this country at least, I think it is quite difficult to see them without being referred by a GP first? I have private healthcare through work and use a private clinic for vaccinations. I have seen a private paed for the conditions mentioned above. All these people referenced the red book (history of birth, weights, vaccination record etc) - GP, paeds (private and NHS) and private clinic for vaccinations. They also updated the records there.
For this reason, I think it is best to stay in the system, get the red book and then just don't go and see them if you don't find the HVs of any use. Round here they only come to your house once after the birth to drop off the red book - rest of the time you go and see them if you choose to, so no imposition really.
One more thing you might want to consider - again depends on your situation, many private paeds also work in the NHS and, as specialists, see very, very sick babies there. They can be a bit scathing when well-off MC Mum turns up with a relatively trivial problem that she gets immediate access to a paed for through private healthcare - I am not sure how yours will take to having appointments booked up with you asking about breastfeeding, teething, weaning etc?