Given that auto dealerships probably make a good deal of their profit
in financing vehicles, is it still true that paying with cash is
Better? It might be. You need to do your homework. With a new car they frequently have a deal 0% or $1,000 off. Sometimes it can be better to get a loan from your bank, and get the cash from the dealer. Though proving you did get $1,000 off the deal is harder to prove.
Other than the obvious no-no of telling the salesman how much
money you brought with you, what are good strategies to employ? Should
you haggle on price or trade-in-value of your car first? Tell them
you're paying cash first?
Normally I would advise you to get a loan commitment from your regular bank. That way you have a offer with a known rate and term. You would only execute that loan if the dealer didn't have a better loan deal. But you are planning on not needing a loan.
Next get an offer from a place like carmax. They will give you an offer good for 7 days. Do this before you are ready to buy a car. That way you know what another place will pay you for your existing car.
Tell them upfront you are not trading in a car. Tell them upfront you will not be needing a loan. This simplifies the negotiating, If you don't tell them this they will be modifying the price, rate, term, and trade-in value. It is easy to become confused. Paying without a loan means there is only one variable.
Regarding cash vs check:
With cash you can be out the door today. If you need a check it can slow down the process. Some places might not take a personal check, they may want a cashiers check. That would mean giving them some money by cash, or check to hold the car, and then making a trip to the bank and returning later.
Cash has risks. How much is needed? You will have two forms submitted to the government: one when you pull the money from the bank, and one when you buy the car. The good news is that they will cancel each other out.
Regarding a check, I have used this extra trip to the dealer to get an offer on my old car, one time they made a better offer so I took it.