Before I get to my answer I think there may be some misconception here that I want to clear up. In Buddhism there are multiple "buddhas". The one we tend to call "the Buddha" (capital B) in the West is Shakyamuni (the historical Buddha) known as Siddhartha Gautama. The buddha you're thinking of is Maitreya, the buddha to come.

It's not unreasonable that you might confuse to the two different buddhas, a lot of people do. Budai, one of the most popular depictions of a buddha, is supposed to be an incarnation of Maitreya in some Buddhist traditions. Your question might be better phrased "why was the image of Budai incorporated into some Buddhist traditions?"

The simple answer is that Buddhism is excellent at incorporating native traditions into its world view and has a monetary incentive to do so. The adoption of Budai probably has nothing to do with racial relatability. It is likely to do with reaching as many adherents as possible. At the end of the day, Buddhism is a money making organization. Whatever it can do to encourage donations, in any form, it will do. More adherents means more people donating to your monasteries.

We can see Buddhism's accommodation of foreign beliefs particularly well in the Vinayas. In a story translated by Gregory Schopen, which you can read in the book Little Buddhas, the Buddha pronounced a rule against treading on cloths: "Monks, strips of cloth must not be tread upon!" Should one tread on a cloth, they would incur an offence. However, he is also said to have pronounced a second ruling: "I, monks, order you to tread on a strip of cloth when asked by a lay person for a ritual of auspiciousness!" While it was "against the rules" to tread on a cloth, monks were actively told to break the rule for a layperson. By performing desired rituals, despite the fact that they may not be "Buddhist" in nature, monks would get more donations.

There are also other deities, Hariti is great example, that were incorporated into Buddhism too. In the Mulasarvastivada Vinaya, Hariti is cast as a child-eating demon. She's eventually confronted by the Buddha who reforms her and turns her into a protector of children instead. We have archaeological evidence that such a deity could be found in Iran and Northern India, so we know she was not a unique Buddhist creation. Buddhism just took the local traditions of a culture and incorporated them into itself. People were already worshiping Hariti as a form of child protection. By adopting local deities a local monastery could bring in more worshipers and, therefore, more money. Amy Langenberg argues this in her paper Pregnant Words.

So, did Buddhism incorporate Budai to make itself more relatable? It's possible. If a story proved popular with the common folk it would be a good idea to adopt it. It encourages donations in all forms: statues, paintings, money, and food. Was it done for purposes of racial relatability though? That's not likely, though is speculative. Speculation is against the rules of this subreddit, so your question might go unanswered here. I'm not sure how I would go about proving a negative. The more compelling evidence I have tells me that many Buddhist figures were adopted for reasons of practicality, often involving money. Buddhism arrived in China around the turn of the millennium. The story of Budai comes about in the 10th century CE. If Budai was necessary for racial relatability, a figure similar to him would have been created earlier. But Buddhism was already doing quite well in China by the time of Budai.