A (hypothetical) US patent on the Turing Machine would have stopped any attempts to build a mechanical device to Turing's Specifications (like this one: http://aturingmachine.com/index.php) for a period of 20 years.

It wouldn't stop anyone from building a device that is computationally equivalent to a Turing Machine; it wouldn't stop anyone from using the abstract notion of a Turing Machine as an intellectual device.

In short, a patent would have had no practical effect at all.

Now, let's imagine another alternative: one where Turing invented the Turing Machine, but chose not to publish it, as he was afraid of some better-financed competitor learning his "trade secrets" and exploiting the device before he could implement it himself. Where would we be then?

Patents are a trade-off. The inventor gets a monopoly for a limited period of time, in exchange for the idea being passed into the public domain after the monopoly expires. It's not a perfect system by any means, but it's not easy to dismiss out of hand.