Most of North America is made up of recent European immigrants who aren't ethnically tied to America at all.
The Central and South Americas have had some immigration but nowhere near on the same scale. A lot of that also happened earlier, so there has been more mixing of cultures. The mixing of the cultures is what creates the name "Mesitzo". That could include one Spanish person marrying a native South American at some point.
Your understanding is basically correct. A complete answer to your question would involve an interdisciplinary review of academic books, papers and Census Bureau statistics. Here's a recent paper: Becoming Hispanic: The negotiation of ethnoracial identity in U.S. census interviews (Latino Studies 2018) by Jennifer Leeman, George Mason University https://www.reddit.com/r/USCensus2020/comments/a4vebg/becoming_hispanic_the_negotiation_of_ethnoracial/
The apparent whiteness of Latin American residents of the U.S. can be viewed as an artifact of the questions the Census Bureau asks about race and ethnicity and the Census Bureau's data processing methodology. Indeed, the Census Bureau says it imputes race for 40 percent of persons who identify as Hispanic and "Some Other Race". See Checking "Some Other Race" Alone -- Self-Affirmation or Political Mistake? https://www.reddit.com/r/USCensus2020/comments/8fznvk/checking_some_other_race_alone_selfaffirmation_or/
I don't quite understand your question. In Brazil the majority of the population identifies as white (48,8%, data from 2010), Chile has a even bigger majority of white people (57,2%). We also have Argentina with 67,3 and Uruguay with 88,8%.
Some people have said about recent European immigration, but if we are talking recent in like 2 centuries, it's no different from the countries I have just quoted.
There's also a whole in this immigration theory. The USA started in 1980 till now to receive more immigrants than ever, but from the 80's till now, the white population started to decrease.
References:
Argentina and Chile race groups: https://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1005602
Brazil census of 2010: https://censo2010.ibge.gov.br/noticias-censo.html?busca=1&id=3&idnoticia=2170&t=censo-2010-numero-catolicos-cai-aumenta-evangelicos-espiritas-sem-religiao&view=noticia
Uruguay race groups: https://www.indexmundi.com/uruguay/demographics_profile.html
Immigration to Brazil: https://brasil500anos.ibge.gov.br/estatisticas-do-povoamento/imigracao-total-periodos-anuais.html
Immigration to USA: https://www.migrationpolicy.org/programs/data-hub/charts/immigrant-population-over-time
White population throughout the years in USA: https://web.archive.org/web/20141114183703/http://www.census.gov/population/www/documentation/twps0056/tab01.pdf
https://factfinder.census.gov/faces/tableservices/jsf/pages/productview.xhtml?src=bkmk