That's not a quote from the paper, that's a mishearing of something he said out loud in an interview. He said "generally caressing" not "genitally caressing".

I can find no source for this statement.

"patriarchy", "privilege/oppression" stuff all seems to be based on a very flawed theoretical underpinning.

The concept of patriarchy or oppression is not unique to the third wave. What do you think is their very flawed theoretical underpinning?

I still think it doesn't stand. I think the "privilege/oppression" thing has obvious counterpoints contained within it

My argument was that the fractures of feminist thought in different camps is not an inconsistency. E.g. that the difference in opinion of a radical feminist and a mainstream feminist are not an inconsistency of abstract feminism. Your argument is that some of these camps may have invalid sets of beliefs. Our arguments are not mutually exclusive. I believe that each feminist camp would agree that the other camps' theories are wrong. Otherwise they wouldn't be in different camps.

is it a privilege for women's diseases to get more research funding than men's diseases?

There are two considerations. Whether the category of privilege is useful and what is the empirical state of affairs. I understand privilege in feminist discource as following. A set of systematic disadvantages (regarding access to political power, economic resources, cultural recognition, legal protection) for one group, based on a part of their identity (e.g. race, gender) is termed oppression, the lack of those disadvantages (as a negative advantage) for the other groups with opposite/different identifiers is a privilege of those groups (in this sense, not in the sense of a formal privilege). Example: Members of group identifying and socially identified as black are more likely to be stopped by policemen when driving (decreased access to legal protection). Members of group identifying and being socially identified as white don't have that disadvantage (normal access to legal protection), which is termed a privilege of the second group in this sense.

I think the category is useful as it is possible in the abstract for systemic disadvantages of some group of people and not others that would be privileged in this sense. There remains the empirical question of what disadvantages actually exist and for whom and why. But regardless of the answer to this, there is no reason to expel the category as it has explanatory power.

If men's diseases are systematically getting less funding than women's diseases and if this means decreased access to some social function and if this happens because of their identity as men, then according to this understanding it would be valid to characterise this a privilege or negative advantage of women. I have not followed literature on intersectionality (how different privileges and identities intersect) so I can't tell whether I am right in my belief that privileges are incommensurable and thus don't even out, so that two groups may be privileged or disprivileged at the same time in different ways without being equal because of that.

explained away with epicycles ("benevolent sexism")

This term is used to refer to sexist attitudes that materialize in an occasion where the sexist agent is at least from his subjective point of view benevolent, e.g. when women are "put on a pedestal". This category is useful because we can conceive of a case where the idealization of an agent may at the same time be structured on top of sexist presuppositions and confining them to presribed behaviours, or that it is coupled with benefits for not objecting to the confinement, or that the idealization or benefits would be used to justify the confinement and its presuppositions. This category can also be applied in cases where mens' idealization is used to confine them to a role. To the extent that this is termed toxic masculinity it seems that we have a problem with too much terminology for the same thing.

On the other hand I can think of subtle differences between the three (toxic masculinity also entails sanctions like the "not a real man" charge and corresponding decreased access to recognition, while benevolent sexism may entail an aspect of lost freedom or power that privilege doesn't). I think a meta-theory would be useful to see to what extent there is overlap between the conceptual content of these terms. Either way, benevolent sexism is not an epicycle and insofar as it is used interchangeably with privilege something is lost of its content.

when other theories do a much, much better job of explaining the reality (men as disposable agents, women as valued non-entities, for instance

Within patriarchy theory there are no non-disposable agents, in the sense that every agent is only instrumentally valued insofar as they can execute the prescribed social role. That may be for them to be house servants or to die in a war. To the extent that they will actualize these roles, they will be valued as good instruments (their value would consist in being efficient housewives or brave defenders etc, and would otherwise be lost, because the state of affairs is not built on neither does it recognise inherent value).

The empirical observation that men are disposable seems trivial (this is the structure/agency problem in sociology. In a sense to say that A has more power in a social system may in certain cases be misleading if we don't add that they have that power as actuators of structured or systemic behaviour -and only insofar as they remain so- and not as free agents with their own perception of the good). I don't know the sense in which you think women are valued non-entities. If you mean to tie their idealization with instrumentalization / lack of agency then I think that's also true. In both cases I see no substantive departure from how I understand terms such as "patriarchy".