Hi Renee!
I suspect you haven't found cover letters for gaining experience because it's not usually the strongest way to position yourself for paid employment.
Are you applying for an open position? If so, they probably need someone with certain skills and/or experience, so highlight that. If you are applying somewhere without an open job req and a small budget, you may have more luck because they are always looking for more hands and it sounds like you're willing to do anything and that doesn't require a high level of skill... i.e., be an unpaid intern. If you'd like to be paid, remember that employers hire for their needs, not yours.
Like you, my primary goals are to gain experience and network, but you won't catch me saying it because: a) it's true at ALL points of my career, b) it's more relevant to employers to know how hiring me supports their goals, and c) I like being paid for what I have to offer.

One place where I might bring up what I want to learn is if it makes your application more relevant without inhibiting your ability to do a great job. For example, maybe there's a composer you want to gain familiarity with because you've done a lot of work by others in his genre. I ALWAYS want the cover letter to say lots more about what I'm bringing to the table.
If you still want to go the "gaining experience" route, you can still highlight attributes about yourself that the employer considers an asset. When I started my first job search, I spent about 2 days painstakingly developing a thoughtful set of "fingerprint" skills that could help me stand out. At that time, I felt I had no marketable skills, so they ended up being personality quirks that I positioned as assets (ta da!!!). That resume was probably only given out about 5 times, but the time was really well spent. I *still* refer to those skills when I'm marketing myself for something I have little experience with (because, guess what, your personality doesn't change a whole lot...). They have been so useful for talking points during interviews, and ironically over time, my work experience has only reinforced the prominence of those skills. The key is to make them really and truly represent who you are because interviewers can tell if they don't mesh with their perceptions of you (or worse, if they don't mesh with your references' perceptions of you).
Also, hiring is different by company. Not all companies have Production Managers. Research when possible. Hiring managers are always impressed by that.
