I would definitely say, to save yourself this worry again, if you have another child you should consider going private - if all the female doctors are busy in theatre/with other women and the only one available is male, I doubt they'll call a doctor away from someone else when your circumstances aren't 'exceptional', if you get what I mean - no declared history of rape or abuse. You'd probably have more luck making this request by going private.
It is tempting to echo that by the time you're in established labour you won't care, because it probably is true, but it's not a helpful statement so I won't bleat on and on about it at you. Personally the only male doctor I saw during the labour was the male anaesthetist who had to come and insert the syntocin drip, but that was only because I have tiny veins, they'd already tried to get the cannula into every vein in both arms (several times) and I'd started developing blockages and clots in the veins; normally your midwife will be able to fit the cannula should you need it (although I think you'll need the anaesthetist if you want an epidural, I'm not sure as I never had one - although the anaesthetist could be female, and as you say, they don't really need to see the business end).
So for the rest of the labour and delivery, I only saw female midwives and doctors, until it came to being stitched up, by which time after they'd been stitching me for an hour and still getting nowhere, they had to call in a senior consultant, who was male - but, despite being very nervous (I'd never had a proper internal examination before labour), by this point I was so off my head on gas and air and total exhilaration and joy at having my little girl, I really really really didn't give a shit and don't remember it now!
There's every chance you could get away with only having female staff, and obviously you can request it - but please be prepared that you may not get it, please don't hold it against the staff as it's a staffing level and procedure issue, not their personal agenda against you, and please don't risk your life or the baby's by going it alone! It really wouldn't be worth it - a moment's discomfort (which you may not even feel uncomfortable about at the time, I distinctly remember apologising to the doctor for not tidying up enough before giving birth, and then asking OH in front of everyone if he'd ever sleep with me again now that it had been shredded; dignity = gone!), forgotten almost instantly, and almost certainly won't bother you after the birth.
Don't forget you can request minimal examinations even if you're being induced. They only examined me once, four hours after breaking my waters.
Best of luck, hope this doesn't come across too harsh or dismissive - I really do sympathise! 