Based on my reading of this question so far, I believe the answer the question lies in whether giving psak is considered a position of serara in our times.
The majority opinion is that a convert may not sit on a Beth Din judging a born Jew (for example: see Rambam in Hilchos Sanhedrin 2:9), though there is a minority opinion that a convert may sit on a Beth Din judging a born Jew for all cases except those in which the death penalty could be applied (see: Rashi on Shabbos 102a). Interestingly, the Tziz Eliezer 19:47 suggests that if the litigants agree in advance that a convert may serve on the Beis Din, then he may serve.
However, as far as I can tell, there does not appear to be any issue with converts becoming rabbis these days (though they could not serve on a Beth Din). I happen to know of several highly regarded semicha programs that admit converts, and Rav Moshe (Iggros Moshe 4:26) even says that a convert may serve as a Rosh Yeshiva, as this is not considered a position of serara in our times.
I was a bit confused by this, as I thought the primary reason that women could not pasken was a question of serara. But this does not appear to apply to converts, as far as I can tell. In the case of women, I believe there is an issue of tznius, but this has always been less clear to me.
This left me with several questions:
Do you believe that a psak is valid if given by a convert with legitimate semicha? If so, then what exactly are the limits of serara what does it not apply to giving psak? What is the critical distinction between a convert serving on a Beth Din for financial disputes and a convert being a congregational rabbi? Would there be a distinction between a convert being a rabbi and a convert being a posek? Is there a reason other than serara that women cannot be rabbis that I was simply unaware of?
There is a difference between being a Rabbi that is answering questions to a judge working for a court in a legal system. This is since a court comes with ways of forcing their ruling (see Bach CH:M 7-1) on those who doesn’t listen to it, therefore being a Judge is considered serara (jurisdiction or authority) and it is different than a ruling on a question in Yorah Deah or Orach Chayim. I think this is what R’ Moshe is referring to in the Teshuva the you mention (Y:D 4-26). Regarding becoming a congregational Rabbi, or just paskening halachic questions on numerous topics will depend on the nature of what the position entails, and has to be discussed on an individual basis. See sources.
Regarding women, the primary reason they are not appointed judges is not because of serara, rather because they cannot be a witness (See Tosefos Nidda 50a D:H Kol Hakasher who brings the Yerushalmi that learn this from a posuk, also see SM:A CH:M 7-4,). With regard to Devorah, Tosefos says that she was a special situation that it was permitted.
The reason why women are not Rabbis or appointed to public positions is for a number of reasons. First of all, because in order to become a qualified Rabbi, the person has to study a lot of Talmud, and Shulchan Aruch in depth, something that women are not supposed to do (See Sotah 20a, Yorah Deah 246-6). Secondly, there is a tznius issue for women to take public position.
Thirdly, there is a serious hashkafic flaw in women becoming Rabbis. Unfortunately, nowadays there are many people, inspired by women’s right groups, the reform and conservative, that are promoting that women should be Rabbis. The core of this, is because they are simply not happy with the way that H-shem made them. Orthodox Judaism believes that H-shem made men different then women, because He divinely sent them to this world to accomplish different jobs. Men cannot bear children, and cannot conceive others. Men are different then women physically, emotionally, hormonally, and psychologically, and anyone who tries to deny this is burying their head on the sand to the world they live in. Each of the sexes have their areas of talent and strength. Orthodox Judaism believes that these differences are because men were given a different job and purpose in this world, therefore each one was given different tools to accomplish their job. It is silly for a skilled successful carpenter to be jealous and want the tools of a dentist, because each profession needs its tools to get its job done. A person that wants to ideologically change this is essentially stating that he or she is not happy with the way that the Almighty’s divine wisdom created him/her but they want to change that, and run the world according to their understanding. Therefore when one person tries to force their way into playing the role of the other sex is stating that they are not happy with the way G-d made it. The position of a Rabbi is to represent H-shem’s Torah, and the Torah’s ideals. Therefore, for a woman to be an official Rabbi is a bit of an oxymoron, because one can fight and represent the will of G-d at the same time with the same position.
Teshuvos Vhanhagos 3 305, Kovetz Bais Levi 15 pg. 131, Mishnas Hager pg. 209.