Both my brother (Nov 1943) and I (Apr 1940) were born at Bronson Memorial Hospital; and I can assure you we were not illegitimate. I don't believe it was a maternity home for unwed mothers (which in today's terminology indicates that they lived there up until the birth of the baby). This hospital was in downtown Renton on the corner of Main & Second Street.. This was the only hospital in Renton until the "government-sponsored Renton General Hospital opened in 1943". That must have been after November 1943, because my brother's birth certificate clearly shows he was born at Bronson. As someone else said, it is a site on the historical walking tour. When I was there in 2014 there was a pawn shop at that location, and it was there in 2010 also.
The hospital that someone on here may be thinking about is the second hospital in Renton -- the "modern" "wheel and spoke" hospital, which apparently the nurses hated because each ward was on a separate "spoke" and they would have to walk back to the center of the "wheel" to get to another ward.
In 1940, my mother was in the hospital ten days after my birth, and my father had to carry her out of the hospital because she was so weak from being in bed all that time (before wheel chairs, I guess). My parents lived in Renton on North 3rd Avenue, so it is logical that my mother went to Bronson. With the information that you have found, I think it is not unusual that the birth mother would have gone to Renton and not to a hospital in Seattle. After all, births were announced in the newspapers and listed by hospital, so people who knew the family, reading the "vital statistics", might not have looked at Bronson Memorial.
Also -- with regard to needing an address in 1940. It wasn't always necessary. I have a "congratulations on the new baby" (me) card with the envelope that is addressed to my parents and the ONLY address on the envelope is "City". That "City" was Renton; which at that time had a population of 4,488 -- four times the population of Bellevue. Remember -- Bellevue wasn't always what it is today. From Seattle, It was either a long drive around Lake Washington or a Ferry ride -- the floating bridge didn't open until July 2, 1940. Bellevue's population in 1940 was 1,177. So it wouldn't be unusual for a letter to be delivered in Bellevue without an address.
Up until the last few years, there were two mail drop slots in post offices -- one was for local mail and one for all other outgoing mail. Mailmen knew their customers; and mail was delivered twice a day, except on Sunday. You need to think in terms of the time period of which you are seeking information. Back then, a letter sent anywhere in the same town would reach its recipient in a day; today it has to go a central processing facility; so it can take three days for a letter to go one block.
You can search for Bronson Memorial Hospital and find a photo of the old hospital. Also, if you are in the area, go to the Renton Historical Museum. They have old newspapers (which are also available on microfilm) and you might find your father's birth in the "vital statistics" section. You have to make an appointment to use the research room at the back of the Museum, but it's worth your time to check out what they have.