People today with a lot of EEF autosomal are much darker skinned than people with a lot of WHG autosomal. By the time EEF invaded south europe around 6000BC WHG were already white and likely more "white" than EEF along with light eyes
Mediterranean people have a lot of EEF while baltic and scandinavian people have a lot of WHG, who looks more white to you?
If you read this you can also say that whg were lighter skinned than yamnaya people because scandinavians are lighter skinned than yamnaya people and biggest difference between scandinavians and other europeans is their higher levels of whg autosomal -
"Yamnayan DNA tested by Haak (2015), Wilde (2014), Mathieson (2015) showed that Yamna people (or at least the few elite samples concerned) had predominantly brown eyes, dark hair, and had a skin colour that was moderately light, lighter than Mesolithic Europeans, but somewhat darker than that of the modern North Europeans"
You really need to do some research. Look up all the papers on the search engine which deal with pigmentation. Then look up the arrival of farmers from Anatolia. It wasn't 6,000 years ago; it was 8,000 + years ago.
So, do you have papers which provide pigmentation data for WHG from that time period showing they carried the snps for "European fair" skin? If you don't then your statement should be given no credence.
So, no, WHG didn't have modern European skin tones.
Nor, for that matter did the steppe people. They were darker than modern Europeans as well. Now, Sintashta type people were more fair, but then they had picked up European farmer genes. If you do your research you'll find more than one paper which proves this.
Now, as to the European farmers, yes, they were probably more fair than the WHG, but most of them did not carry the derived copy of 45A2, which was selected for later. Nor did the CHG people carry it.
There were some Anatolian farmers who carried both the major light skin genes and light eyes, one even added light hair to it. However, something happened evolutionarily in Europe where people of mixed ancestry wound up with the two major light skin genes of the farmers, the blue eyes more common in WHG, and light hair. Climate, specifically latitude, definitely has a something to do with it, as we've known for a long time, i.e. in more northern latitudes it's harder to get Vitamin D, although if you eat a lot of fish and organ meats you can get it that way. In today's Europe those in more northern or more cloudy areas (Ireland) have the palest skin. Light eyes usually, although not always, go along with it. Blonde hair need not be included, i.e. the Irish again.
All of this is explained in great detail in the many papers written about the subject.
It's called evolution. If you're going to study genetics of any kind it's important to understand it.
If you think about it logically, you'll realize that the WHG fled into the north and far northeast to get away from the farmers, or to islands far in the west, the very areas where there is less sunlight. So, it's not that the WHG alleles for pigmentation CAUSED the lightening of skin, but that the MIXED people who had a decent amount of WHG happened to live in areas of low sunlight. Correlation is not causation.
Btw, Sardinians have quite a bit of WHG above the amount in the EEF, and no one would call them fair.
Now, before you argue the matter further, it would be better if you would read the darn papers.