Should fixed assets be reported on an income statement ?

No. Assets belong on the Balance Sheet.

Start with your accounting equation:
Assets = Liabilities + Equity

That is also your Balance Sheet. This is the "financial position" of the company.

Revenues and Expenses are your two other kinds of accounts, and those actually belong under Equity in the end. But we keep track of all those temporaily in their own accounts and they are summarized on the Income Statement. So the Income Statement is only revenues and expenses.

The net income (or loss) from that statement is applied to the equity, and then the updated equity is used for the Balance Sheet.

Fixed Asset - Balance Sheet