Wizards only rarely breed with muggles

They do breed with each other happily enough.

Wizards have a fairly high mortality due to for example the wizarding war and various dark lords.

I highly doubt any dark lord was worse than the Tatar invasion, the colonization of America or the 20th century, or that the wizarding world has had more or more bloodier wars.

No person among hundreds dying in school is considered an achievement, compared to a muggle school Hogwarts has a bad safety record.

I'll concede that one, but I don't think that's a demographic-scale problem. Even a 50% school death rate should be offset by wizard families being fertile twice as long. And you can always not send your kids to schools where they die violently half the time.

There is a tendency to get fewer children for developed nations, I would expect the same to hold for wizards. Japan

Japan in particular has issues that won't translate to the wizarding world. Wiki says:

One cause is the traditional Shinto belief that overpopulation in Japan will upset the Kamis and natural balance. Buddhist teachings are also similar. A number of factors contributed to the trend toward small families: high education, devotion to raising healthy children, late marriage, increased participation of women in the labor force, small living spaces, education about the problems of overpopulation, and the high costs of child care and education.

I'd expect magic and House Elves to negate most of these.

Also:

In an article in the August 2009 issue of Nature, Myrskyla, Kohler and Francesco Billari show that the previously negative relationship between national wealth (as measured by the Human Development Index (HDI)) and birth rates has become J-shaped. Development promotes fertility decline at low and medium HDI levels, but advanced HDI promotes a rebound in fertility.[14] In many countries with very high levels of development fertility rates are now approaching two children per woman — although there are exceptions, notably Germany, Italy, Japan, although Japan is showing signs of rebounding fertility rates.[15]

Kindly elaborate as to how muggles existing is strong evidence (aka rules out any possibility of any other theories) for a population cap among wizards being in effect

• If wizards are human, they would have produced an exponential population growth at least on par with the rest of humankind until overpopulation becomes an issue. That's what humans do, that's what they've always done, and that's what I expect from any human population.
• If the wizarding gene also makes you want or be able to have less children, I consider that a mechanism of the magical limits, not a separate theory. There's also the Weasleys.
• The only other explanation I can think of is that wizards are not actually human, in which case they couldn't interbreed with Muggles.