...does the number of emergency cases cost more than the total number of private procedures carried out?...
This is immaterial.
When procedures are carried out by the NHS they generally occur in facilities equipped to cope with complications, and are often dealt with before they become (vastly more expensive to deal with) emergencies. On the other hand, when procedures are carried out in private facilities they rarely have the ability to deal with complications, resulting in a reliance upon calling 999, shifting the burden of a problem caused by private healthcare (and the costs of dealing with it) onto the publically-funded NHS.
...If there wasn't private healthcare would those staff still be required...
Yes. At present there are approximately 90,000 vacancies across the NHS, about 40,000 in nursing roles alone.
...which would likely be an additional cost to the NHS if they were...
But it'd likely be a net saving, taking into account less need to use Agency, Locum, Bank and OT staff to cover vacant shifts.
...Does the NHS not benefit from those staff's abilities during the hours that they are working for them?...
I would argue it's the other way round. Very few specialist staff work predominately in private healthcare, with a bit of NHS on the side. In my experience for the vast majority the NHS is their steady paycheck and main workplace with private healthcare being a bit on the side.
Not only is this because private healthcare can be a fickle industry, with workload coming and going but, as already mentioned, staff need to stay competent in their various roles and it's the NHS who largely funds such training. Few private healthcare providers give anything other than very basic training, with most competencies 'read across' from their NHS posts.
To add to this the vast majority of cutting edge technologies are tested within the NHS and for good reason. As an organisation there is far more institutional knowledge available to assist with introducing such procedures and more importantly, the NHS is self-insuring. Few private healthcare providers on the UK are willing to contemplate introducing new technologies before the NHS has ironed out any issues.
...the point still stands that because the private healthcare is a benefit it results in more tax being paid.
As it should.
Private healthcare is not compulsory, it is a service like many others consumers can purchase. Should I get a tax break because I eat healthy and exercise regularly and so am, in theory, less of a 'burden' on the NHS? Of course not.
Paying for the NHS also isn't just for the 'now', it's for the future. What if a few months from now you lose your job and potentially access to private healthcare? Should you receive a lower level of NHS care because you previous took a tax break based on being "less of a burden" at the time?