I do think it's partially a British problem. I grew up in continental Europe loving veggies, never gave it a second thought. As a child, it wasn't "eat your greens" or "a portion of veg" or "five a day," it was just integrated into the everyday diet. I'm not saying fussy eaters don't exist in other cultures - my sister was one - but I wasn't aware there is such a "battleground" around vegetables until I lived in the US and here.
I'm not a fussy eater at all and will eat literally anything - canned soup, boiled cabbage, offal, raw fish, I'll have it all. I'm also quite militant about clearing my plate. But OMG, when I tried boiled peas and carrots in a British restaurant... it was honestly the first time in my life that I ever wanted to spit my food out. I still shudder thinking about it. Some restaurants are better than others of course, but the standard in your typical pub is pretty bad tbh. So I really can't imagine kids having much fun with it either if they get similar at home.
I have several British friends who are much, much better chefs than I ever will be - I don't think culinary skills are the problem. My theory is that the quality of veg in the UK is generally not that great. The vegetables in the supermarket all look freakishly perfect but ultimately taste like cardboard. Not sure if this is due to the climate, commercialization of farms, or maybe due to strict standards in supermarkets!?
I find I eat a lot less veg in the UK myself as it's just not enjoyable, but I load up when I go back home. I always take an extra suitcase and come back to the UK with bell peppers, potatoes, arugula, tomatoes, you name it... I get some weird looks at security but it's worth it!