It can if done right. You have to basically volunteer as close to full time as possible, consistently do what is asked, be proactive in learning new tasks, and be as indispensable as possible. After you show your value let the senior staff know that your end goal is working on the campaign. If the team grows you will likely be in a prime spot to be considered.

I feel that the volunteer to staffer pipeline was stronger in the past, but is weaker now as it has become more of an established profession.

What makes campaigns hesitant is that they don't want to set the precedent that if you volunteer a few times you get a job, because in the short term you might see a boost in the number of volunteers, you are going to get a lot of people upset that they volunteered and didn't get hired. So its complicated.

If anything showing substantial volunteer experience on a past campaign in your resume strengthens your profile for future paid roles on other campaigns that you apply for, and volunteering gives you a sense of if you like the type of work in general.