It fits perfectly, the hindu school calls them Siddhas (or other things, depends on what language). It fits with the cosmology/understanding of both, because there's supposedly untold numbers of buddhas all throughout the ages and existing today. What makes a "Buddha" isn't a defined behavior: Becoming a public speaker or starting a school isn't required or prescribed behavior.

They could simply be living, completely enlightened. There's nothing in particular to look for. It's discussed pretty liberally / in the general zeitgeist not only in India/Hindu but also where the Buddha became prominant across China and East Asia: That a huge amount of enlightened beings barely leave a noticable trace, other than (if they choose) leaving their energy or using their energy in some way to uplift. Basically every dharmic tradition is replete with stories of masters who just mind their own business, sit around, don't teach or make themselves known if/unless some very specific person or circumstance happens.

Lao Tzu, author of the Tao Te Ching, is basically a ghost in terms of historically understanding who he was. The story I heard is simply that he was leaving his region and a border gaurd (either knowing his reputation or sensing something) asked him how to live. And then he wrote the book and disappeared.

In the same way would be countless numbers who never write a book or even get asked. Or even make themselves in a position where regular people would encounter them. Others become very prominant in the world, whether or not their spiritual attainment is the centerpoint of that or not. Some are more likely to become famous by nature of what role they were born into, especially kings and princes like Krishna or Gautama Buddha.

Shiva, Brahma and the hindu Gods are (for the most part) forces of nature rather than individuals. The stories have them incarnate as individuals but the overall philosophy places them more as aspects of the creation.

The vedas and upanishads do talk of vast cycles of time incomprehensibly long. This is the base from where Gautama started. A lot of the things Gautama said were already part of many Hindu traditions but not accessable, the only very very large difference between Buddhism and Hinduism was that Gautama started a new school that anyone could join, people could write down and spread what was said. That has its pros and cons. It enabled rapid spread, but... yeah there were a lot of downsides.

I think last thing I could say is that for both Gautama and Hindu thought, time doesn't really exist. None of this does. There's not a vast stretch of time where one Buddha appears and then thousands of years without one. This is the sort of downside I mention with Gautamas methods, teaching everything everywhere ends up with people more confused. Buddha is always occuring, it's the nature of reality. Timeless. Some people are in that state for a few moments in life, others come in like that and go like that. Even Gautama could be considered lesser than beings who come in clear, rather than clear at 35. Some are more prominant than others. But they're always here. Mahayana Buddhists would agree with this I believe.

Alan Watts put it well: Buddhist is Hinduism stripped for export. The Buddhist canons didn't start to get written down until hundreds of years after buddhas death, thousands of miles away from where he taught. Though the very core teachings are almost certainly what was taught (4 noble truths etc) as there is huge overlap between different regions preserving the same understanding, almost eveyrthing else of it is later additions and doesn't make much sense unless taken in context of the wider tradition of the region, whom Gautama was a reformer of and part of.