There is no reason to worry. The worst thing which can happen is that a piece of bay leaf, being somewhat hard, can lodge somewhere in your digestive system, necessitating a trip to ER. But a medical paper on the topic starts its discussion section with the sentence "Reports discussing ingestion of bay leaves have been exceedingly scant". They only cite 10 references in the period 1950-1990, and most of these are general studies of foreign bodies in the esophagus, not specific studies of bay leaf ingestion.
Given how often bay leafs must find their ways into people's digestive systems (they feature in our food), it is safe to conclude that only a tiny fraction of ingested bay leafes cause problems, else there would be more studies mentioning such cases. The same is true for side effects different from mechanical obstruction: if this had happened, somebody would have published it.
The paper I mentioned is "Bay Leaf Impaction in the Esophagus and Hypopharynx" by Stephen K. Buto, MD; Tat-Kin Tsang, MD; Gerald W. Sielaff, MD; Laurie L. Gutstein, MD; and Mick S. Meiselman, MD. Sadly, it isn't freely available (I could read the full text because my uni has a subscription).
I guess that if you are working as a cook, your workplace may decide that even if the chance for a customer choking on a bay leaf is something like one in a million, they'd rather instill removing bay leaves from dishes as a policy. Probably prudent, although there are more important risks to care about.
I'm not sure if there is a scientific explanation for this. I also think it's more that vanilla enhances the overall flavor profile of the dish rather than actually bringing out other flavors. Salt, on the other hand, does enhance flavors.
Best Answer
Dried bay is sometimes left in, for example this herb mix from Schwartz includes bay leaves (at less than 3%).
Dried they can easily be crushed and added as a powder, but you don't often want a lot of dried bay as the flavour can be quite overpowering.
I only use fresh bay as I have a plant outside the kitchen door and it's evergreen. Fresh, I have been known to get them fine enough to leave them in, but it's much less effort to just use them whole, perhaps slightly folded or torn (without tearing them into pieces). To leave them in you really need to make a paste or of them, using a blender/food processor (except you generally wouldn't be using enough for them to blend well). Pieces of fresh bay leaf are rather tough and prone to getting stuck in teeth, but fresh bay has a better flavour than dried.
Unlike many herbs, you don't have to put in any extra effort to prepare them for removal - they're tough and obvious, so no tying sprigs or little bags.