I'm a doctor and a lot of patients often ask about the best way to stay healthy. Its a complex topic and one which people just often feel overwhelmed by, which is a main reason why obesity is soaring - people just feel like they cant understand healthy eating!
I think what you're talking about as 'healthy carbs' are ones with a low glycemic index (GI). Glycemic index is a measure of how quickly the carbs you consume affect your blood sugar levels. A high GI carb will cause a quick sharp rise in blood sugar whereas a low GI carb will cause a slower more gradual increase that doesnt reach the same 'peak' high as higher GI carbs.
At the end of your day sugar is sugar to your body. Low GI foods just have this sugar packed up more tightly and it takes your body more time to break the food down and absorb the sugar. This is important for diabetics who want avoid that blood sugar peak but also for everyone else.
Erratic blood sugars and high GI food will eventually lead to insulin resistance and the multitude of ill effects that has on your health.

At the end of the day its everything in moderation. Eat a good amount of fruits, veg, proteins, and also some fats. If you like puddings that fine, have the occasional one just not 5 in a day.

For anyone interested here's a like ranking some food from low to high GI index. https://www.weightlossresources.co.uk/diet/gi_diet/glycaemic_index_tables.htm