I've had blood drawn and x-rays taken without anesthesia, however my a-vet does prefer anesthetizing birds for x-rays because you need the bird to keep still for the pictures to be taken. It usually requires putting birds on their backs and sides for x-rays, and this freaks out most birds, hence anesthesia.
As far as blood goes... well, my a-vet takes from the jugular vein.... she typically bends the needle before inserting into the vein, as she finds it easier to draw blood. Once the blood is taken, they release the neck of the bird, and the way that birds normally sit, it stops the blood from flowing. Here are some videos as examples.
Some avian vets feel more comfortable going the anesthesia route because it is less stressful on the bird when they are out cold to take blood and do other routine examinations. If the vet doesn't feel confident in using anesthesia or not using it (considering his current health and their comfortability with those procedures), you may need to drive further to find another avian vet.
There is Ian D Hunter at Galemire Veterinary Hospital about 1 1/2 hours away from you.....
Welcome to Galemire Veterinary Hospital
Or Andrew Greenwood at International Zoo Veterinary Group just slightly further away....
International Zoo Veterinary Group, Avian and Exotic Pet Clinic Facilities
Found those two via this link.... --->
Member Search Results - Association of Avian Veterinarians
Found some other links that may be helpful in finding an avian vet?
Avian Vets | Parrot Veterinary Services
Birdline UK Parrot Rescue - Vet Listings
Avian & Exotic Referrals | Services | VetIndex
Avian Vets in Yorkshire & North East England
Although I would recommend calling these places first to confirm there are avian vets at these locations, then seeing if you could speak with an avian vet about your situation before making an appointment. Many vets over in the USA don't always like talking to customers over the phone as they may be able to get more information in person, so you might have some trouble there....