I'll take hypothesis. I don't see an overwhelming amount of evidence though. That is more of an opinionated term.
That's the opinion of experts in the relevant fields of biology.
Koonin and Wolf (2010)
Conclusion
A formal demonstration of the Universal Common Ancestry hypothesis has not been achieved and is unlikely to be feasible in principle. Nevertheless, the evidence in support of this hypothesis provided by comparative genomics is overwhelming.
Larget et at Statistical evidence for common ancestry: New tests of universal ancestry (2016)
Abstract
While there is no doubt among evolutionary biologists that all living species, or merely all living species within a particular group (e.g., animals), share descent from a common ancestor, formal statistical methods for evaluating common ancestry from aligned DNA sequence data have received criticism. One primary criticism is that prior methods take sequence similarity as evidence for common ancestry while ignoring other potential biological causes of similarity, such as functional constraints. We present a new statistical framework to test separate ancestry versus common ancestry that avoids this pitfall. We illustrate the efficacy of our approach using a recently published large molecular alignment to examine common ancestry of all primates (including humans). We find overwhelming evidence against separate ancestry and in favor of common ancestry for orders and families of primates. We also find overwhelming evidence that humans share a common ancestor with other primate species.
Yonezawa and Hasegawa Some Problems in Proving the Existence of the Universal Common Ancestor of Life on Earth (2012)
Abstract:
Although overwhelming circumstantial evidence supports the existence of the universal common ancestor of all extant life on Earth, it is still an open question whether the universal common ancestor existed or not.
the 2 I remember most were the peppered moths and the stages of birth. The updating of those texts books is a well known issue that schools and texts books were not concerned with.
The pepper moth experiment has been replicated and proven to be correct - see this paper based on a six year study conducted between 2002 and 2008, which replicated the results of the original study.
Selective bird predation on the peppered moth: the last experiment of Michael Majerus
So, no the peppered moth is not false, or a deliberate forgery, or a hoax.
The same with the "stages of birth" - by which you have to mean Ernst Haeckel's embryonic drawings. Creationists like to claim that they were falsified, when in reality Haeckel was just interpolating because he lacked sufficient information.
Here's the NCSE's take on it.
Yeah we have had to develop new mechanisms because Darwins theory doesn't have what it takes. When it one naturalistic theory fails, we hypothesize other naturalistic theories, like punctuated equilibrium. That is the process of methodological naturalism.
Punctuated equilibrium is a mechanism within the evolutionary framework. It was initially assumed that evolution was a gradual process (phyletic gradualism) that occurred more or less at a steady rate. Gould et al showed that instead, evolution occurs in fits and starts, with periods of explosively rapid (from a geological timeframe) and widespread morphological diversification (cladogenesis), then periods of comparatively slower or more concentrated diversification.
By legitimate theory you mean a naturalistic theory. God is not natural but supernatural. I don't share your assumption of metaphysical naturalism.
Science is necessarily limited to methodological naturalism, ergo any theory that replaces evolution would have to also be limited to the realm of the natural.
If you want to overturn it, you have to provide an explanation of biological diversity that better fits the available evidence - not religious dogma.