What exactly are you trusting a cert provider with and what are the security implications?

End users trust the cert provider. The cert provider has a process that they use to determine if they can trust you.

What attack vectors do you open yourself up to when trusting a certificate authority with your websites' certificates?

You’re not really trusting them with your certificates. You don’t give them your private key or anything like that, and the certs are visible to anyone navigating to your website.

Your new vulnerabilities are basically limited to what you do for them - any changes you make to your domain’s DNS config, or anything you host, etc. - and depend on that introducing a vulnerability of its own. You also open a new phishing attack vector, where someone might contact you, posing as the certificate authority, and ask you to make a change that would introduce a vulnerability.

In what way could it benefit security and/or privacy to utilize a paid service?

For most use cases, as far as I know, it doesn’t.

LetsEncrypt doesn’t offer EV or OV certificates, which you may need for your use case. However, these are mostly relevant at the enterprise level. Maybe you have a storefront and want an EV cert?

LetsEncrypt also only offers community support, and if you set something up wrong you could be less secure.

Other CAs may offer services that enhance privacy and security, as well, like scanning your site to confirm your config is sound… but the core offering isn’t really going to be different (aside from LE having intentionally short renewal periods), and theoretically you could get those same services from a different vendor.