Why do Chinese grocers advise cutting and discarding flowers in Chinese vegetables?

When a plant arrives in the kitchen, the ecological perspective doesn't matter any more. Any insect present on a plant destined for human consumption is considered a pest by the consuming humans and by the cooks preparing the food for them, no matter what the plant considers it to be (pollinator, parasite, whatever). The average person dislikes consuming insects, and people who enjoy eating insect-based dishes frequently object to consuming insects of unknown species that are not part of the recipe. This is a cultural thing of course, my own great-grandfather always laughed at people cutting out worms from apples - for him, the worm was as much a part of the apple as the pips. But the average westernized cultural setting today views that insects are disgusting and should not be eaten.

So the seller won't tell you which insects these are - they don't know and it doesn't matter to them. They just assume that you don't want to eat the insects, like the overwhelming majority of their customers. And so they tell you what method to apply - cut away the flowers - to avoid eating the insects. I suspect that they may have had outraged customers returning vegetables as "defective" if they found out they have insects in their food, and have taken to preemptively give advice how to get rid of the insects.

As for the objective harm, that's highly unlikely to happen. There are very few insects which can damage your health, and these tend to be human or animal parasites, not the kind of insect which lives on a vegetable. So if you eat the flowers and swallow the insects unnoticed, no harm done. If you notice them, there can be emotional harm or not, depending on whether this is a disgust trigger for you or not.

I would say there may be more insects in the flowers, but that is not a big deal. The ones not removed by normal washing or possibly a saltwater rinse will mostly at least be tiny ones attracted by the flowers and will not affect quality and taste. Some would even callously call it extra protein. IMO, the real issue with these types of plants having flowers is that the plant is bolted. Most if not all of these vegetables are best quality when harvested before they bloom. Any bolting would normally mean the plant was beyond prime condition when harvested or was grown under stress, too hot, too cold, too crowded, not enough water are typical causes of early bolting. The entire plant may be fine, but sometimes will be tougher or bitter and the flowers and stem on many are bitter and tough and not what you are looking for. They often will have a very different taste than the rest of the plant. In some cases, you might like that taste, but in most it is not what you were expecting. For instance, in Pak Choi I have had it with flowering. Not only was the stem to the flowers stringy, it had a latex like with liquid which was off-putting. The flower itself had a strong mustard taste which was not at all like the leaf and stem which is what was intended.

I believe that this advice is strictly your local Chinese grocer, and does not apply in general or anywhere else.

I have shopped at many different Chinese, Vietnamese, and Korean grocery stores in several different US cities. I have Chinese friends that I cook with. I have never been personally advised to remove the flowers before.

Further, I checked several online guides on preparing gai lan for cooking, including this, this, and this, and none of them advised removing the flowers. Nor did the books Chinese Greens nor Fuschia Dunlop's cookbooks.

So, speaking in general, they are not correct (which makes the second part of the question irrelevant).

However, that particular Chinese grocer may be correctly speaking about that specific batch of gai lan that they are selling. So maybe check those (or go to a different grocer).