You’re not responsible for code you haven’t pushed. This seems like a bad client that not only wants to push code cheaply but plus wants you to be responsible for it. Charge extra. Your value is not in generating code but the accountability of the code you generate and verify.

In my experience, you are "responsible", regardless. It doesn't matter who wrote the code originally.

Have you worked in a true client services model? Based on your statement, my guess is you have not. This is NOT how things work at a consultancy or in a client services model, regardless of industry, unfortunately.