I'll give you my story and maybe it will apply to you. I started cycling about 5 years ago in Greenville, SC. All we did was climb hills - all the time. I didn't race and most of our group rides were rides to destinations where you regrouped and then road the second half of the ride. A lot of climbing - a lot of elevation gain. For fun, we would do long rides of 10,000 ft of elevation gain. I was fit and in what I considered very decent shape.
Two years later, I drug my bike up to MD to visit my parents and ride the Seagull Century (or something like that) - the ride was flat as a pancake. And I crushed it. I don't remember by average MPH but it was the fastest I had ever ridden. It was a start when you feel like it ride & I started later (due to sleeping in) and I just passed people all day long - even tandem pulling pacelines - just cruised right by them. I had a guy who flatted and his friends / group left him. He jumped on my wheel and I drug him back up to his friends - he thank me a bunch and even bought me a beer after it was over. In summary, I felt like "the man" for sure. I was convinced that since I trained in mountains that it made me a "strong" rider.
6 months after that I got transferred to Orlando FL. Packed up my stuff and moved down. I spent the next 6 months getting dropped from all the fast group rides in town. Confused - I couldn't understand it. I thought I was a "strong" rider, but I guess I was strong when I managed my cadence and my pace. These guys just seem to hammer and hammer away and I always seemed to get dropped at some point on the ride. Eventually, I gained some weight and built up the muscles required to keep up with those fast paced rides & learned how to hide in the wind a little bit.
So my point is - climbing may give you a good base to go fast on the flats but depending on what you are trying to do on the flats - you would be better to train for that specifically.