Using WACC as discount rate for project

Yes, you are correct that the statement is fairly generic, and not necessarily specific to the project in question. If a company finances a project entirely with debt, its debt to equity ratio will rise, its debt cost may change (if the credit rating goes down) and the WACC may alter. For this reason companies often use a "optimal" cost of capital for appraising projects, which is based on the optimal debt/equity ratio. If the actual debt/equity ratio is not currently at that level, the blame is placed on the treasury department, and not on the department running the project. This gives at least a consistent WACC for all projects.

It is not true that debt costs would always rise with a new project. If a company finances a new project by raising capital in the same proportions to the company's current capital structure, the debt cost and the debt rating should not change. Yes, the amount of debt will rise, but at the same time the assets in the company will rise (and those assets cover the repayment of the debt, and earn the profits that pay interest on the debt). Nothing should change in this case. Additional funds do not necessarily bring additional financial risks for the company.