Philosophy has few consensus positions, so "what philosophers think" will almost always be a scattershot answer.
The question you asked, is based on applying falsification testing to a claim -- basically extending Popper's methodological naturalism to philosophy. Popper endorsed this, but not all philosophers do.
ENOUGH do, however, for these sorts of test cases to have convinced most philosophers today that essentialism is doomed. And pulling on this thread -- may take you were most other philosophers have been reluctant to go.
The basic issue is that virtually nothing in our universe appears to have an essence. The Sorites paradox, about a pile of whatever, and the Ship of Theseus thought problem, illustrate that all objects we identify in our world are bundle objects. They remain "that object" for us, even when they lose some parts, or add other parts, or even have all of their parts replaced, bit by bit. I.E. -- objects do not reduce, and they cannot be identified uniquely.
This creates a major problem for physical reductionists, and for analytic philosophy in general.
For physical reductionists, the pragmatic physical reality, that originally convinced them of physicalism, basically has to be denied. This denial is called scientific realism, and it holds that there really ARE no higher level objects, the REAL realty is just the most fundamental of physics. We just identify higher level objects as a mental shorthand, since dealing with the reduced reality of our world is just too much work. There is in this view ultimately no hardness, solidity, picking things up, etc. That philosophers are reluctant to accept the unreality of our experienced world has been an ongoing challenge for reductionists. This has led most physicalists today to accept emergent physicalism, where higher level objects ARE real, as they emerge through some as yet TBD process for lower level structures.
However, the way those higher level objects CHANGE over time provides a massive challenge to analytic philosophy. If my car is in an accident, and has many body panels replaced, plus I decided to upgrade the engine as it was also damaged, and unreliable to start with, and I repainted it another color -- is this now the same car? Most of us treat it as the same, but this seems -- odd. Trying to reduce identity to its VIN number also does not work, as I could pry its VIN off and bolt on one from a junker. This would be illegal, but would not plausibly change my car into the old junker.
The problem for analytics is that it seems that, moment to moment, due to molecule losses and changes, and larger pieces being replaced, A=/=A for basically any object in our universe over time. And if A=/=A, then one cannot validly apply logic to it.
Enter essentialism. To be able to do logic on objects, if some essence is assumed for that object, then analytics can still be done. As Conifold's comment asserts above, many essentialists have tried to assert essences are reduced properties. These are things like DNA for dogs, or for a particular person -- or the VIN number for a car. However, DNA as a category does not work 100% of the time, as all animals go thru a single strand -- double strand life cycle, and sometimes members of a species have more or fewer strands, etc. And as you noted, the DNA criteria does not separate dogs from wolves, as they interbreed. Another classic essentialist reductionist argument was that water must be H2O, in all worlds, based on laws of physics. BUT -- physics laws change, and minor variations in the values of the Standard Model of Quantum Mechanics in a different world would lead H2O to have different properties from what we call water. These efforts to identify an essence through reduction -- all fail due to the bundle problem of reality. Bundled objects and categories are GENERALLY true, but every one has exceptions, and that includes the efforts to identify a reductionist justification for essences.
Personality/selfhood essentialists have a similar problem. As Hume noted, our selves are bundles. More recent thinking no longer identifies selfhood with the highly variable and fleeting experiences and thoughts that Hume identified as the self, and instead focus on more stable memories and inclinations (character). But while selfhood is not at variable as Hume asserted, our memories are added, and fade, and are sometimes wrong, and our inclinations change. So even a more stable version of self faces and essence problem. Spiritual dualists have an answer to this -- asserting an essence of "soul". BUT -- this leads to a massive complexity in spiritual dualism, as soul was initially introduced to solve the problem of consciousness. But if consciousness is a variable bundle, and souls are not, then souls don't help at all with consciousness! And the relation between an unchanging soul essence and self/agency/consciousness becomes a murky quagmire, which depending on the relation proposed, will face its own falsification tests.
Where reductionists go from there -- I don't know. Most I have read just look for confirmation from the "generally true" of their claimed reduced essence, and they don't look for the more exotic falsification tests. Denial of falsification testing, or else making the claim untestable in principle are the general reactions among the ideologically committed to seeing their ideas falsified.
Aside -- I noted that essentialism is needed for analytics to stay relevant to our world, and that this thread can lead you far from most contemporary philosophy. There is another alternative to analytics, which need A=A, and for truth to be absolute. The alternative is radical pragmatism, where truth is what is useful. Under radical pragmatism, analytics are recognized as invalid per their own criteria when applied to this world, but often highly useful anyway. This question of A=/=A, among others such as the Munchausen trilemma, the inability of any language to support analytics (see Two Dogmas of Empiricism), and the plurality of logic, have lead me to radical pragmatism.