Did ancient and/or medieval cultures that emphasized the danger of religious pollution also exert more control over women?

The anthropologist Mary Douglas's Purity and Danger (1966) is a foundational text on pollution and taboos which won't lay your question to rest—and neither will this answer—but does present a model of pollution taboos consistent with the hypothesis that cultures with strongly held beliefs regarding purity are likely to find more reason to police the bodies and behaviour of women. At the risk of oversimplifying and leaning too much on this one aging but still-seminal book:

Before germ theory defined 'dirt' as that which is infectious, dirt was "matter out of place."¹

[Dirt] implies two conditions: a set of ordered relations and a contravention of that order. Dirt then, is never a unique, isolated event. Where there is dirt there is system. Dirt is the by-product of a systematic ordering and classification of matter, in so far as ordering involves rejecting inappropriate elements. This idea of dirt takes us straight into the field of symbolism and promises a link-up with more obviously symbolic systems of purity.²

In cultures with strong taboos, this 'system' is a thick worldview in which dirt symbolically implicates other fears and desires:

Any structure of ideas is vulnerable at its margins. We should expect the orifices of the body to symbolise its specially vulnerable points. Matter issuing from them is marginal stuff of the most obvious kind. Spittle, blood, milk, urine, faeces or tears by simply issuing forth have traversed the boundary of the body. So also have bodily parings, skin, nail, hair clippings and sweat. The mistake is to treat bodily margins in isolation from all other margins.³

So the physical realities of menstruation, childbirth, and breastfeeding render women's bodies more marginal (lined with margins) than men's, and thus more likely to be sites of boundary trangression. (Recall too that until very recently and still outside the rich world, high rates of infant mortality mean that childbirth is much chancier to begin with, inviting the investment of ritual significance.)

By Doulgas's account, it is the perception of danger which generates taboos which (symbolically) address that danger; and critically to your question, the fears exercised are contingent on the circumstances of the culture in question:

There is no reason to assume any primacy for the individual’s attitude to his own bodily and emotional experience, any more than for his cultural and social experience. This is the clue which explains the unevenness with which different aspects of the body are treated in the rituals of the world. In some, menstrual pollution is feared as a lethal danger; in others not at all […]. […]

Each culture has its own special risks and problems. To which particular bodily margins its beliefs attribute power depends on what situation the body is mirroring. It seems that our deepest fears and desires take expression with a kind of witty aptness. To understand bodily pollution we should try to argue back from the known dangers of society to the known selection of bodily themes and try to recognise what appositeness is there.⁴

For instance, in the way promiscuity is regarded within the frame of the Hindu caste system:

Since place in the hierarchy of purity is biologically transmitted, sexual behaviour is important for preserving the purity of caste. For this reason, in higher castes, boundary pollution focusses particularly on sexuality. The caste membership of an individual is determined by his mother, for though she may have married into a higher caste, her children take their caste from her. Therefore women are the gates of entry to the caste. Female purity is carefully guarded and a woman who is known to have had sexual intercourse with a man of lower caste is brutally punished. Male sexual purity does not carry this responsibility. Hence male promiscuity is a lighter matter.⁵

Sadly, there are of course many causes of misogyny and the disenfranchis*m*nt and control of women besides notions of purity (if causation even runs that way at all); and there are causes for pollution taboos besides the symbolic/metonymic implication of remote or closely related fears and desires. Corpses, for instance, don't just transgress a metaphysical boundary between life and death, but also stink—because we've evolved to recognise their smell as disgusting, because they're infectious. So while there certainly is a relationship between pollution taboos and patriarchy, it's very messy and contingent, as you seem to have suspected.

It seems reasonable to guess that the relationship between pollution taboos and patriarchy will be stronger and more straightforward the more you narrow the scope of the taboos under consideration to those which address margins associated with women but not men in society, such as menstruation taboos.

Finally, the contingency of taboos on the unique set of dangers perceived by a culture might go some way towards explaining, to speak to one of your examples, why even in ancient Greece, across which a roughly consistent pantheon was worshipped, Spartan women enjoyed considerably more rights than their counterparts in democratic Athens—despite the Spartans' famous superstitiousness/religious probity.

Douglas, Mary. Purity and Danger. London: Routledge, 2002 [1966].

1. p44
2. p44
3. p150
4. p150
5. p155