What's the origin of the legend that this rock formation in America had a 12th century Welsh Origin?

The Madoc story was discussed at length in The History of America, published in the late 1700s. The author cites several earlier works for the origin of the story, and you may be able to track it down further from there:

Evans Early America:Umich.edu

Most probably this is good old plain racism and motivated reasoning:

When 'discovered' by European settlers — in the 18th and early 19th century — such artifacts of the 'mound builders' clearly show some form of 'advanced civilization' and sophistication. So, surely, nothing the barbarous Indians could have ever accomplished. Some European nobleman just must have been responsible for such works, first ascribed to him at the Alabama mound that was thought to resemble Dolwyddelan Castle.

(— George Catlin: "Letters and Notes on the North American Indians", (Michael MacDonald Mooney, ed), Clarkson N. Potter: New York, (1842) 1975. archive.org)

Another example for this racist theory:

Modern investigations and discoveries show that there once existed an almost unbroken system of defences, extending from New York, Pennsylvania, Virginia, and the Carolinas, in a diagonal direction, to the valley of the Ohio, and thence into the great basin of the Mississippi. These works increase in size and number as they advance towards the centre, and may properly be classified into forts for defence and tumuli or mounds for sepulture. They are chiefly found along the fertile valleys through which run large rivers, and at their junctions with one another. It is quite usual with writers on these remarkable works to assign to them so great an antiquity that the employment of figures is almost useless if they tell the truth.

But there are substantial reasons for the belief that they were erected by the Welsh, aided by those Indians with whom they became incorporated and whom they directed in their labor.

The route they took, either by choice or necessity, and the exact correspondence of these earthen monuments with those found in England and Europe known to be of Cambrian origin, go very far to support this belief.

— Benjamin Franklin Bowen: "America discovered by the Welsh in 1170 A.D.", J.B. Lippincott: Philadelphia, 1876. p71, archive.org

That the superficial resemblance of the Alabama mound is based primarily on a tower at the Welsh castle that was only erected in the 13th century, after Madoc had supposedly left? Nothing to ruin a good story.

Is this a simple hero tale? Or did these intrepid settlers reach the New Land and stayed? The land that Madoc settled is believed by some to be Mobile Bay, Alabama. Today a plaque stands on the shore to commemorate the voyage. But there is more than just empty tales to spark the curious. Along the Alabama River are a series of pre-Columbian “forts.” Cherokee tradition, as reported by Governor John Seiver of Tennessee in a letter dated 1810, states that the forts “had been made by the White People who had formerly inhabited the country…” The ancient structures, like so many others along the Atlantic Coast, were unlike any others constructed by Native Americans. One of these forts, located on the summit of Lookout Mountain, is said to be “nearly identical in setting, layout, and method of construction, to Dolwyddelan Castle in Gwynedd, North Wales.” Gwynedd is allegedly the birthplace of Prince Madoc. According to archaeologists, the forts were constructed around 1100 AD but are believed to have a “religious” purpose. These Welsh settlers reportedly traveled up the Alabama River were many of the stone forts were constructed. In 1799 Governor Seiver reported that six skeletons had been discovered in the area clad in brass armour and bearing the Welsh coat of arms.

Another Native American tribe, the Mandan, apparently were believed to be remnants of the Welsh settlers. Very light skinned and bearded, the Mandan used coracles, a round boat very similar to those still used in Wales today by fishermen. Mandan villages were laid out in squares with streets also similar to European design. The most striking thing about he Mandan was their language. Evidently, it was so similar to Welsh that they easily understood Welsh spoken by an American lieutenant who was from Flintshire, North Wales. Unfortunately, the Mandan tribe was practically exterminated by smallpox in 1837. A few Mandan reside today on the Sioux reservation in North Dakota.

— Gary R. Varner: "Ancient Footprints. Cultural Diffusion in Pre-Columbian America", OakChylde, 2010. (p32–33, academia.edu)

— Jerald Fritzinger: "Pre-Columbian Trans-Oceanic Contact", Lulu, 2016. (gBooks)

For the origin after 1583 of the Madoc story itself:

— Matthew Lauzon: "Welsh Indians and savage Scots: History, antiquarianism, and Indian languages in 18th-century Britain", History of European Ideas 34, 2008, p250–269. doi