I have mixed feelings about this proposal:
On one hand, the term "Caucasian" is indeed questionable and so are "Native American" and "Hispanic". There's no reason to connect white people specifically with the Caucasus, Amerindians are not "native" to the Americas - it's an attribute used for no other group - and "Hispanic" covers way too many very different and unrelated population groups.
The term "European", on the other hand, is perfectly good, as it designates people with very distinct phenotypes and identifiable genetic clusters that developed in Europe over tens of thousands of years, just as Southeast Asians, Northeast Asian, Subsaharan African etc.
Yes, each of those groups are divided into numerous and very diverse sub-groups, again with very distinct genetic clusters that allow their identification.
Subsaharn African populations have the largest genetic variations - but are also at the greatest genetic distance from all the other racial groups.
These populations obviously had very little contact with each other over tens of thousands of years, maybe significantly more.
First Humans Arrived in North America 116,000 Years Earlier than Thought: Evidence from Cerutti Mastodon Site
http://www.sci-news.com/archaeology/cerutti-mastodon-site-humans-north-america-04815.html
It's obvious that each population group changed significantly over tens to hundreds of thousands of years of separation.
The simple fact is: we can easily identify members of each of those groups, they do have distinct genetic clusters that allows their identification at the genetic level and in a study where people were asked to self-identify with one of the major groups (European, Southeast Asian, Subsaharan African Amerindian etc.), their self-identification matched the identifiable genetic clusters.
What's more important is that these racial groups also have very specific medical needs, as medication developed for one group may not work for the others. e.g. medication for heart conditions developed for people of European origin were found to not work well for people of Subsaharan African ancestry - and cardiac problems are more frequent among them.
To quote just 2 examples from medical research:
Racial Differences in the Association of Serum 25-Hydroxyvitamin D Concentration With Coronary Heart Disease Events
https://jamanetwork.com/journals/jama/fullarticle/1710459
The Relationship of Age, Race, and Ethnicity with Survival in Dialysis Patients
https://cjasn.asnjournals.org/content/8/6/953.abstract?sid=67afc60d-8092-42aa-860d-1865a47ef6c3
There are other inconvenient studies that confirm that genetic differences are significant and do result in issues that should not be ignored:
Prematurity and Low Birth Weight as Potential Mediators of Higher Stillbirth Risk in Mixed Black/White Race Couples
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2867623/
Following the mass migration from 3rd world countries into Germany in 2015, the media were forced to report that they had to constitute specific blood reserves for Africans, as European blood is apparently not compatible.
NRW-Forscher suchen Blutspenden für Migranten
https://www1.wdr.de/nachrichten/rheinland/blutspender-mit-migrationshintergrund-gesucht-100.html
To quote: "Doch europäisches Blut führt bei Migranten und Flüchtlingen aus Afrika oder dem arabischen Raum häufig zu Abstoßungserscheinungen mit schweren gesundheitlichen Folgen, bis hin zum Tod"
Translation: "European blood causes rejection in migrants and refugees from Africa and the Arab region with severe health consequences - potentially including death".
So this classification is actually useful and not at all superficial or "racist".
The geneticists who call for such changes in terminology are either too focused on technicalities or they want to be politically correct, which is a much more likely motivation than a desire for scientific inaccuracy.