Would it be dumb to buy a brand new car with cash using half of my savings?

I think it would. Buy a car with under 1,000 miles on the clock and about 1 month old and you will save thousands. Dealers have courtesy cars - buy them  

No.  I always pay cash.  Borrowing money means you have to pay interest.

Sometimes the interest rates are so low it is better to finance. Keeps you more solvent and since you have enough in the bank to cover the price you will have enough for payments should something unexpected happen with your job.

How much do you make?  How much is the car?   How stable is your job?  Will you get fired if your company starts having difficulties?  How much money do you have after all the bills are paid?  The answers will tell you if you if it is dumb or not.

A rule of thumb is you need to have money to cover 3-6 months of living expenses just in case things do not go right.  Think about that and make your decision.