As is often the case on this board, my view is different. If you have maintenance which has not been done, yes, you need to catch up. You might need that car a while.
However, assuming you change the oil with some regularity, it is unlikely you will have any problems in a 3,000 mile trip – again assuming you drive near the posted speed limits, and don’t run 90 to 120 mph.
Traditionally, most highway failures have been tires; belts; and hoses. At legal speeds, there is no more probability of that car failing in 3,000 highway miles than in the last 3,000 miles. Highway miles are easier on a car than driving around your home area.
My 2002 Sienna runs an easy 2700 rpm at 70 mph. Not hard on motor at all.
With over 190,000 miles on it, last October we left home here in Mexico, and drove 830 miles to the border. Then, 3 days and 1500 miles to Florida. A week later, 1500 miles back to border, then after preparing, another 830 miles home. Total, um, over 4600 miles. No problems at all, except I chose to install new motor belts which had well over 130,000 miles on them. No failure, my option. I also changed the oil and filter, of course.
I took my first cross country trip in December 1964, 2050 miles in a 1953 Chevrolet. 50 hours to Ft. Lewis. I have done many cross country trips over the years, and the only real problem I had was something I induced once and I am not going to talk about it, heh, heh.
You need to think about other things. Make sure you have a cell phone. If you don’t perhaps Straight Talk from Wal-mart with a month card for it. Take several gallons of water, they sell that stuff in gallon jugs. In the summer, if the car did die, you could be suffering from thirst. This is basic survival techniques and does not reflect on your car. We always took a large container of peanut butter. With that and water you can survive a rather long time. And, since you will be prepared for an emergency the odds of having one are slim.