How does the reaction against gender-specific pronouns relate to a languages' use of gender?

That's not an unreasonable hypothesis but it when you look at the details, the gendered morphology of a language only plays a partial role in how the language expresses underlying cultural gender bias.

1. In languages that require constant expression to gender in verbal or adjectival agreement (like Russian or Spanish), you're unlikely to come up against the issue of the epicene pronoun as you do in English. Because the pronoun does not refer to an implicit assumption about a default gender but to the agreement with the referent. So in Czech, a sequence like. "If a person likes something, he/she will buy it." will use the feminine pronoun because the word 'person' is feminine. However, you will find that men will actively avoid trying to refer to themselves in female gender. E.g. with sentences like "I am the person who said..." men would try to find different ways of saying this without having to end with a sentence that has 'I' to be the subject of a sentence with a feminine conjugation.

2. But the issue of gendered language is not really about the pronoun. The pronoun reflects the default framing of a man being in a position of power and access to resources. So for example, the assumption that a doctor is male and therefore a sequence like "A: I went to see a doctor. B: What did he say?" reflects a male bias in the framing of the term doctor. Many languages require that professions morphologically mark gender. Therefore, in the above sequence you'd always have to say if you went to see a male or female doctor. However, this does not remove the issue of gender bias. Because in sentences like 'I should see some kind of doctor.' or 'All doctors are rich.' The default (unmarked) gender will be male. So if you said something like 'All doctors[fem] are rich.' You would be in fact stating that all women doctors (as opposed to men doctors) are rich.

3. There are many other ways in which these languages express bias through defaulting to masculine gender. For example, in Czech the past tense plural agreement is male by default unless you are talking about an all female group. All it takes is one man in a group and the agreement switches to male. Another example in Czech is the use of feminine nouns when referring to men in a negative way.

4. The ubiquitous marking of gender causes all sorts of other issues with which the speakers of these languages have to deal on a daily basis. For example, if the language (like Czech or Russian) expresses gender in the verbal agreement, a woman talking to man will have to work harder to switch between the genders when talking about hypotheticals.

5. Because of the nature of gender and morphological agreement, many of the solutions proposed for English would be difficult to implement. For example, in English you can deal with the default assumption of gender relatively simply. For example: "The doctor will need to be good if he or she wants to succeed." But in a language with rich gender morphology, you would have to say something like: "The doctor[fem] or doctor[masc] will need to be good[fem] or good[masc] if he or she wants to be successful[fem] or successful[masc]. So the imposition on the speaker is much greater.

So there are many factors that come into play when trying to explain why languages like these do not lend themselves to reforms designed to even out the gender prejudice similar to those in English.

But morphology and noun gender are not the only (and perhaps not even the main) reason why many of the cultures which speak these kinds of languages do not deal with the issue. The reasons are primarily cultural, historical and political. For example an easy reform for many Slavic languages would be to stop inflecting women's surnames with -ova (which implies possession by the male - e.g. Nováková implies implies Novák's). There are very many good linguistic tools for a language like Russian or Czech to deal with nouns without gender marking by simply not inflecting them. Yet, nobody has advocated that and when I tried to at least not put the -ova morpheme at the end of foreign female names of my translations, my (female) editor insisted on putting them there. The fact that there's little appetite for radical or even small scale action is due to the lack of a history of feminist movements dealing with language and also active anti-feminist campaigns (often supported by women) in these countries. The issue is complex and far outside the scope of this SE. However, it does speak to the close linkage between politics and claims about the 'nature' of this language or another.