Ancient White Americans:
When discussions regarding the potential of Whites inhabiting ancient America, the following are the prominent topics:
—Solutreans, Northeast NA.
—Kennewick Man, Washington State.
—Windover Archeological Site, Florida.
—Blond haired peoples, South America.
—Viking settlements, Northeast NA.
—Phoenician voyages.
—Roman voyages.

The Viking settlements were few, small and abandoned as quick as they started, leaving no lasting impact beyond some medieval ruins in Vinland (Canada). Greenland settlement was significant however if you count that.

Theories of large-scale Phoenician and Roman (and even Viking) voyages to the New World are common and widespread, however likely not true. There's no DNA traces of any of these groups, if they came to the Americas, they were few, and either left or died out, leaving no lasting impact. With the potential exception of acting as a temporary minority ruling elite, producing architecture, and likely founding Mesoamerican-Script (I would argue this is the case).

Windover and Kennewick archeological sites, which appeared rather European skeletally, were "proven to be actually Amerindian" using fallacious points. That point being ignoring ANE ancestry.

Ancient North Eurasians:
A significant portion (40-60%) of Amerindian DNA (ancient and modern) comes from an ancient Caucasian group called ANE (Ancient North Eurasians), a group modern Europeans have a significant amount of DNA from.

The ANE that traveled across and settled the Americas, would have already been significantly mixed with East Asians, but to varying degrees, some significantly purer than others (perhaps many genuinely pure), but they traveled with lots of East Asians and hybridised with them to produce modern Amerindians.

What you should note is that ANE are the origin of blond hair, so many of the mixed peoples that settled the Americas would have the genes responsible for blond hair, and many likely expressed this gene (especially the purer ANE individuals).

Considering this, it is very likely the Spanish accounts of finding blond hair peoples, and elite, were accurate, they would however be mixed regardless. Possibly an elite built on keeping their ancestry as pure as possible to continue expressing this blond gene (what ancient Egyptian elite did with redhead).

It's however clear what caused the gene to die out, the numerous accounts of Amerindians sacrificing blond haired peoples...

With the Kennewick man, he was 47% White (ANE), and likely maintained a more Caucasian appearance due to recent mixing. I can't imagine the story is much different for the Windover site (don't have public access to DNA to check for ourselves).

The Solutreans however aren't ANE, and descend from ancient French/ Spanish EEMH (Cro Magnon). These peoples migrated to North America from Europe and were never mixed, as there were no other peoples already in the Americas at the time, and migrated from a mono-ethnic region (Europe), unlike the multi-ethnic East Siberia.

Solutreans unfortunately were ethnically replaced and absorbed by the latter ANE-mixed Asians. Outside of haplogroups in modern Amerindians, Solutreans left little genetic impact; whatever caused them to get wiped out so consistently, as to leave little genetic impact on modern Amerindians, is up for debate, and is likely caused by remaining small hunter-gatherer groups, killed by major natural disasters.

Summary:
White influence and settlement in the Americas has always been small and minimal, until Christopher Columbus.
I'd argue their impressive architectural feats (impressive for them, eh for us), and the invention of their alphabet, were the fault of White travelers, or a White ruling elite.
But for the most part, they've always been mixed hapas, living in mud-huts, elevated by their significant (40-60%) White (ANE) admixture.

We all get excited about American pre-history, but for the most part nothing much happened, what we should get excited over is that Whites conquered it all and built a White civilization that the world awes over.