I found that running benefited my hill climbing out of the saddle, but not general cycling. I only started running over winter as I only had a “best” road bike that I didn’t like to use in the rain.
I proved this to myself as I went all out on a 5k park run (21 mins 😆 ), knackering my legs to the point it was painful to walk. But then got on a bike and rode 40 miles without any discomfort other than hill climbing (out of the saddle).
I started running by jogging for a mile, it would hurt for a couple of days, then I’d go out and do it again. After 2 or 3 times I was able to jog for 5k easily. Then I just started increasing the pace, 10k came quite easily after that.
Unfortunately I can’t run at the moment because of a stomach issue, well at the pace I want to, but I now have an all weather road bike. Good luck!