Should arguments based on the proponent's gender be used in a grant proposal?

This does not feel like something you as an external grant reviewer should really care about. The gender of the applicant may or may not be a tie breaker (or even a major factor) for the jury, but they will know which it is. If gender is a factor in the evaluation, they will also be acutely aware that the applicant is female, because either the system or an administrative person will have flagged the application as coming from a female applicant. No need to tell the jury that this is indeed a female applicant.

In consequence, I think you should simply ignore this part of the application. Keep in mind that as a reviewer, you are not deciding on the application - you are merely laying down the facts (e.g., related to the scientific quality of the proposal and applicant) so that the jury can decide based on their own strategy, criteria and availability of funding.

Side remark: note that even should you disagree with the notion that the gender of the applicant should play a role, you should resist "mentally subtracting some points" because the applicant even brought it up. It's not your place to decide if gender should or should not be part of the evaluation. If you would decide to judge the proposal more harshly to counteract a perceived undue advantage, you yourself quickly become part of the problem. Nothing good can come out of that.

The answer invariably lies in the instructions the funding institution provides. If instructions have not been provided, and you don't know how to appropriately review -- as with any other review issue, contact the program officer, or whoever is managing the review process, and ask.

I haven't seen processes that use this information for general science grants, but such issues surrounding diversity efforts are often fair criteria in things like training grants (e.g., the NSF GRFP review process: https://www.nsfgrfp.org/applicants/application_components/merit_review_criteria), so it wouldn't surprise me if there were situations where such considerations are important.

I think there are situations where this information is relevant.

First, as you stated, there could be explicit notes in the call which signal that one of the goals of the funding is to increase the diversity of the field.

Second, you might be asked to evaluate the "promise" of the applicant, or the potential of the applicant to significantly impact the field. This might be common in a "young investigator" program. In this case, I think information about an applicant's individual characteristics and their ability to impact the field as a member of a diverse group might be relevant (in addition to the project's impact on the field of course).

Third, the evaluation guidelines might include some criteria looking for the impact of the proposal on a broader set of institutional goals. For instance, we have a faculty research fund at my school, and one of the criteria is "impact of the project on the university." In this case, if one of the university goals is better representation, then funding this project would advance that. I basically think of this as a PR criteria.

In other situations, where you are only asked to evaluate the intellectual merits of the project itself, this information would not be relevant.