You don't need a support car to go up an Alp
Farewell Francis, you dweeb. Just in case you've confused anyone I thought I'd share the easy shame-free way to do what you've done. (Believe it or not there are cyclists out there who think self-supported touring is nigh on impossible.) It's so simple and straightforward and fan-bloody-tastic that everyone should just get on and do it, often. It shouldn't be a 'once in a lifetime' high drama with a personal mechanic, spare bikes and ice baths. It's just a holiday for fuck's sake. And much more fun than going up an Alp with thousands of other people on a sportive.
Use whatever bike you've got. You don't need a rack and panniers - a bar bag and a biggish tail pack (e.g the Carradice SQR Tour) are plenty. Pack as little as possible. One set of cycling clothes is enough - just rinse them every evening to get the sweat out, wring them in a rolled up towel and they'll dry overnight. For street clothes all you need is one shirt, one pair of ultralight travel trousers and some flip-flops. Stay in youth hostels. French ones are cheap and fun and plentiful and there's no age limit.
Get to France by ferry and by the time you see the mountains you should be fit enough to get up them. Most people don't need a triple chainset, unless they've got panniers full of camping gear and spare clothes. If you pack light you'll manage with 39 x 26. If you can't, just leave your bags at the bottom of the climb in a bar/under a bush. Then you'll have the joy of ascending/descending without luggage, which is bloody lovely.
Most of the big name climbs aren't steep, just very long. Average gradient is typically 7% with the odd stretch of 10 or 12%. So just fucking do it. Alpe d'Huez, the Galibier, Mont Ventoux, they're all just sitting there waiting for you, and there's no hallowed tarmac which is a "privilege to share with the pros". They're just public roads, with a few cars on, which is good because the natives are friendly and will give you water if you wave an empty bottle at them. Lots of them will treat you like royalty just because you're on a bike. The only cars which cut you up are the ones with GB plates. The worst problem you will encounter is increased fury at the way cyclists are treated when you come back.