Does a baby need a recognizable face?

Yes. A logo or wordmark, or both. I don't make a big distinction between the two because a wordmark without a logo is usually strongly stylized so it is recognizable as "not regular written text."

So the answer is yes UNLESS (big exception here) you are talking about a business that is purely local and has inbound demand through word of mouth that is more than enough to use up its supply capacity.

Typically businesses that fit that description have a charismatic one-person owner who is out front being the face and voice, so you don't need a logo because that person's face is enough. Like the Soup Nazi on Seinfeld. You can call such brands implicit brands, with the person's face being the implicit logo. Instead of a face, a distinctive part of the service can also replace the logo (example, a roadside peanut vendor who sells his peanuts in distinctive paper cones that look different from the normal ones of nearby vendors).

But in general, an abstract and explicit logo is required for most normal businesses.

I am not including here the idea of a "personal brand" (either celebrity or social media varieties). I don't like that term because it overloads the concept of brand and dilutes it to the point of vacuity.

The deeper reason is that visual symbols need to be recognizable at a pre-literate level. There are illiterate people out there who can nevertheless recognize famous brands like Coke, because the logo/wordmark is so distinctive and stands out so clearly from regular written text.