> This might be a silly question, but do you think that asking to be engaged for no pay, emphasizing you're very determined to gain experience would generally be a good idea?

Doesn't hurt to ask. A good approach is to see if the lab has a list of research grants on their website. Look at those, and you might be able to infer where funding is available. Same goes for the recent papers, they give you good indications. It's even better if you're taking a class and can engage the team via that.

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>I am in no position to try and aim for these places, it does make me worry that I would have to compete with an abundance of people with such qualifications at every step. I've recently burned myself trying hard to get chosen for summer research programs.

Don't be discouraged, you only need to be accepted once :D

To be fair, 75% your success in applications to Faang and Ivy League will be via recommendations and network. You have a full 2 years to build that.