If the offending material "resides" on a service providers network, they have to remove access to it. If that material is on your PC, access to your PC has to be removed.
This is a basic misunderstanding of the DMCA. Information on your very own PC in your own home does not reside on your ISPs network, it resides on your network. Your ISP is providing services covered under 17 USC 512(a), not 17 USC 512(c). 17 USC 512(a) does not have a takedown provision.
See also the Verizon v. RIAA case, where the RIAA attempted to force ISPs to disclose P2P users identities under another provision which only applies to 512(c) service providers and not 512(a) service providers.
"...any notice to an ISP concerning its activity as a mere conduit does not satisfy the condition of Section 512(c)(3)(A)(iii) and is therefore ineffective."