I enjoyed the pictures and history that you posted...I thought you, and others, might enjoy a few more pictures (Compliments to Mr. Splashbar for the scans. Didn't have scans of 2 cards that were pertinant to your post...so apologies for the poor photos of those two ) . Hopefully I have all the dates right ...

This shot, which includes the Apache Casino, is from 1935.

The Apache Casino closed in 1941, and was replaced by the Western Casino. In 1942 The Western Casino was replaced by the New Western Casino.

The Eldorado c. 1947-1951

Joe W. Brown's Horseshoe

This card is date-coded 1960

I agree that your postcard (artist rendering) makes it look like the Horseshoe building took up the whole block, but the building really only took up 1/2 the block until after the Mint closed in 1989.

FWIW, we have the same postcard, with the artist rendering of the new Horseshoe, that you have. Ours is postmarked 1963.

As you may know, Curt Teich postcards are date-coded and the codes are easy to translate. The date-code is often on the front of the linen cards. There's a 2-character date-code, followed by another number. The date code is a number (year), followed by a letter (decade). e.g. Your Apache Hotel postcard number starts 4A, which translates to 1934 . (so easy to read, so hard to 'splain! )
A=1930
B=1940
C=1950
D=1960
Sometimes the date-codes are on the back of the card, sometimes even in the stamp box.

The Mission Bar, seen across from the Apache in the card below, was open from 1938-1939.