You might want to find a new doctor. Running out of those kinds of meds WILL create a medical emergency and you’d think any doctor would know that and would figure out some way to fit you in even if it’s just a one-minute look at you and hand you the scrip. Methadone withdrawal is nasty and withdrawal from some kinds of anti-anxiety meds can result in life-threatening seizures.
Part 1 is true. Part 2? Guess what, lots of people have legitimate chronic pain problems and are on continuous pain killers and take their medications as prescribed for pain relief. THEY are not the reason there are such strict regulations on narcotics. I’d love to hear your “alternate solutions” (not really) for pain relief. I’ve heard them all and until you’re suffering chronic pain, I’d suggest sticking them in your ear and watch your judgements.
To address the OP: If this happens again, I would suggest calling/stopping in and asking for a 5 day Rx for the prescriptions he needs- that way they know he will keep his appointment and he won’t be without.
Not all narcotic painkillers require a new written script every month. I believe Class IV controlled substances aren’t under this requirement. My Tramadol prescription (Tramadol is Class IV) is just phoned in from the doctor’s office.
Tramadol is relatively easy to get, and you might be able to talk a doctor into an over the phone transitional script for it. (My dog takes it 
My dog also takes Keppra - so our human pharmacy has several prescriptions for my dog)
When we ran out of my daughters ADHD meds, the clinic sent in a prescription for two weeks - provided we had an appointment in that time period. Her doctor also does the “write three prescriptions” and I’d lost count of the months.
Anti anxiety meds are another one that’s tough to get - my doctor hands them to me FIVE AT A TIME - and expects the five to last all month - or longer. But she’ll give me all the Vistaril I want (ok, I’m exaggerating), so that may be a good transitional prescription.