There is no debate
The so-called Extended Evolutionary Synthesis (EES) is just a new term for a whole lot of knowledge that roughly speaking everyone acknowledge to be true. EES is really just an extension but does not say anything that conflict with previous theory. There is no conflict and no contradiction.
The only debate stand upon whether or not our current understanding of evolutionary processes deserve a new name or shall we just stick with what we call the modern synthesis.
Shall we use the term ESS?
I talked about this renaming with about 20 evolutionary biologists and all of them consider such renaming useless. However, by the end of the day, they all mainly did not care much.
My personal opinion about the renaming
IMO, one reason such naming is silly is that there has not been a major work that have completely and suddenly changed our understanding. We just keep discovering more and more as we always did. All the new things we discover we just used to integrate it what we already called modern synthesis. Those advocating for a renaming define the modern synthesis not as most people do (that is everything we know about evolution so far) but as the set of knowledge that existed in 1950.
It is funny that even the discovery of DNA did not bring about a renaming of our theory of evolution as we only considered that it added up to our current understanding. I doubt that the emergence of new fields (such as niche construction and its heritability or developmental noise) would be in any way sufficiently revolutionary to cause a renaming. And I say so even though I also work in the field of mutational and developmental noise.
The points listed in the link are rather unclear and it is therefore impossible to comment on them. Take the #2 you are interested in
heritable variation will be unbiased
The term variation has no strict definition. The term variance is clearly defined. However, a variance in itself cannot be biased, only the distribution it comes from can. So, this simple sentence mainly makes no sense (to me at least). If it does make sense to those who wrote this list, then, OP, you will need to provide explanations from other source or ask for its meaning before asking for its validity.
This paper will offer you opinions from both sides. As implicit from the above, one side (namely, the "counterpoint" side as referred to in the paper) seems to me to represent the vast majority of evolutionary biologists.
Addressing your comments
Thank you for the clarification. I think by "heritable variation," the authors mean mutation. You feel it's still true that they are unbiased?
"mutation" is not a variable, it does not follow a distribution. It cannot be bias or unbias.
Mutational effect on fitness is a variable and follows some distribution which is most definitely biased (deleterious mutations are much more common than beneficial mutations; usually modelled with a double gamma distribution; see here).
Mutational effect on the phenotype is a variable and follows some distribution. It may well be biased indeed. The distribution of mutational effects on the phenotype were mainly unconsidered by evolutionary biologists who contributed to place the first bricks in the modern synthesis (Fisher, Wright, Haldane, Muller, etc...).
These "early" evolutionary biologist never made predictions about the distribution of mutational effects on the phenotype that would contradict today's predictions. They just had little understanding and did not talk about it much.
Again, in your question you are opposing EES with Modern Synthesis as if they were to be opposed. But by doing so, you need to clearly define Modern Synthesis. By default, one just consider the Modern Synthesis to include all the knowledge we have in evolutionary biology and therefore by definition, elements like niche construction, biased mutational effects of phenotype and other field of knowledge that one might want to specifically associate to EES only.
Do you still feel it's true that natural selection is the only source of the evolution of adaptation?
This question depends only on the definition of adaptation. If, for example, you consider a lineage being poorly adaptive becoming adaptive as the environment changes to match their phenotype (and not the other way around) to be an adaptation, then typically nobody would have never thought that only natural selection would cause adaptation. I would personally be happy to call that an adaptation.
The question is not a question of evolutionary process but only a question of the definition of adaptation.
Note, as an extra source of confusion, when talking about adaptation one may refer to the changes yielding to higher fitness (whether the change is genetic or not is up to you) and/or to the resulting state of having a high fitness.
For example, if there is an epigenetic modification that is transgenerationally stable and adaptive, is it an adaptation?
Again, it depends upon what you call adaptation. I would be happy to call that an adaptation.
the discovery of DNA would not cause a reconceptualization of the modern synthesis because it was consistent with it.
Everything from the EES is consistent with the modern synthesis. Most of it are just add-up subjects that brought little consideration before. Some of it might break some simplified generalization that may have been done before though. Many does affect our view of the grandeur and complexity of evolutionary process but none is contradicting what we previously thought.
what you mean by "becoming more adaptive as the environment changes." Would you give an example?
Imagine a population is perfectly adapted to live at an environment where the temperature is 18˚C. The current temperature is 15˚C though. Let's imagine a case where the population does not change much through time (eventually due to a lack of genetic variance to select from, eventually due to a low mutational variance or a recent bottleneck) but the environment is getting warmer and after some time the environment is at 18˚C. The population is now perfectly adapted.