Why there is no "的" after beijing ?
The 的 would be used if the food was from Beijing (like how you would say that restaurant's food as 那家饭馆的菜), but since it is talking about Beijing food in general there is no need
Why is 很 needed here?
Like other sentences, 很 can be used figuratively as "to be", though it literally means "very"
I used really for emphasis and it was accepted too.
Why is 是 not needed?
https://youtu.be/pY5JOMZOWLA?t=23
我想北京菜很好吃 should be acceptable. think = 想! 觉得 = feel like
我觉得北京菜是好吃 Why can't you use 是 in place of 很?
You can, if you add 的 to the adjective. 我觉得北京菜是好吃的。