"is the thought that there was some sort of patent protection that did not allow S&W to chamber their revolver in .45 Colt, thus the new .45 S&W caliber."
No, not really.
The simple explanation is that the S&W No. 3 frame was too small to accept the full length .45 Colt round, and Smith & Wesson had neither the time, inclination, or production facilities to redesign the gun with a larger frame.
Even had the frame been of a suitable length, the design of the .45 Colt cartridge itself wasn't suitable for use in S&W's auto-ejecting revolvers -- the rim on the .45 Colt cartridge was simply too small and would slip under the ejector star, tying up the revolver and making it hard to nearly impossible for someone on horseback to clear the jam.
In order to get their foot into the door, S&W simply designed a new, but very similar, cartridge more suitable to use in its revolver. The case was shorter, and thus held a bit less powder, and the rim was bigger, making the auto-ejection feature work.
The development of the Schofield is kind of an odd duck for S&W.
They had huge ongoing contracts with Russia and other nations at the time, and didn't have much spare production capacity to devote to large US orders (which never came).
In fact, in the late 1870s, the US Government asked for, IIRC, another 20,000 Schofield revolvers, and S&W turned the contract down because they were literally out of production capacity.
"The gift which I am sending you is called a dog, and is in fact the most precious and valuable possession of mankind" -Theodorus Gaza
Baby Jesus cries when the fat redneck doesn't have military-grade firepower.