You can have a similar epiphany with regard to corporate taxes, which are ultimately paid by the consumer.
Think about the two extremes: If taxes are 100% - the business goes under and everyone loses their jobs, suppliers lose a customer, customers lose a product they liked and shareholders lose everything.
If taxes are 0%, the business grows - shares become more valuable, the demand (and therefore price) for employees and suppliers increases, and customers will pay less.
The naivety of your second paragraph suggests you think business is not operating in its own best interest. How does employee and supplier costs going up = consumer cost going down? There seem to be huge leaps being made here.
Corporate tax policy and supply/demand economics is probably too complex to try to boil down to a a short paragraph each.