My hill training has led me to start exploring interval training. Yes, going up a slope is a good way to get good at going up a slope. But I've also noticed - before the gradient gets too steep - that a lot of time on a hill is spent at the top end of your aerobic performance.
You're breathing hard and only just "within" yourself. This isn't a leg thing. Your legs are fine. It's more about the supply of energy and oxygen to them that is being taxed. You need to get better at supplying all they need to power up the relatively easy 8 to 12% slopes that a continental climb is likely to present. And to be able to keep on supplying all they need for 30 to 60 and may be even 90 minutes.
So that's perhaps 90mins of "top end" aerobic performance. To start trying to improve this you can schedule in a "Steady State Interval" - the name isn't quite right to me - perhaps "top end aerobic interval" would better describe it? Go pretty fast over a 10 to 15km flat route - so something that will take you 15 to 25 minutes to complete.
The interval should be quite hard - but not sprint or all out - and probably a 7 or 8 out of 10 on a scale of "perceived effort". Because of the duration it almost inevitably becomes just about the fastest way to cover 15km of flat road - and still be able to ride on immediately after the interval (rather than collapse). You should feel able to say a few words/communicate a bit whilst doing the interval but not carry on a conversation or even say a whole sentence.
This sort of interval will start to reflect the demands you are going to face on a continental sportive where 30mins is nothing for a climb...