Have you checked your minute volume?
Get a bucket and turn all other water sources off. Go to the valve nearest the meter and check the water pressure with a pressure gauge. You can pick it up at the hardware store. It should tell you how many pounds per square inch of water you have. It should be over 40 lbs but less than 60.
Get a watch with a second hand or turn on the timer on the phone. Open the faucet all the way and time the filling for 1 minute. Measure how many gallons of water you fill in a minute. At 40 psi a 1/2 inch 25 ft hose should fill at a rate of 24 gallons per minute. A 100 ft hose will fill at 6 gallons per minute.
You said the tuna cans only collected 1/3 of an inch. Did you put a series of cans out or just one can? To do the can test I would place several can from the sprinkler a few feet apart but in a line. Sprinklers do not lay down water evenly so you need to water for 15 minutes and measure the volume of each can and get the average of the set. If you only get 1/3 inch in 15 minutes you would have to water 1 hr and 20 minutes to lay down a inch.
How much and how often you have to water depends on your soil type and how well it drains. Heavy clay soils will hold on to water longer and sandy soils will dry out faster
Instead of looking at the inches, I would get a trowel and dig into the soil after watering and see how far the water penetrated. If it got at least 4 inches deep, I would be happy. How often to water, dig down every day or two and feel how damp the earth is. When it is almost dry you need to water again.
Usually, unless the soil is compacted, I run my sprinkler 45 minutes twice a week. My water pressure sucks it is 37 psi and I set the water sprinkler to turn on at 2 am since there won't be many people turning on the faucets at that time. While it is not great to water at night, plants get used to it. By the time all my zones are done it will be almost daylight.
Deep watering is better than shallow watering. Roots will go deeper and be more drought resistant. Shallow watering = shallow roots that dry out faster.
Grass is a heavy feeder and in my yard I have reduced grass to a minimum. I do need to renovate my lawn since it has a thick layer of thatch and a lot of weeds. I have cut down on feeding it, since it rewards me by requiring me to mow more. I don't feel sorry for the grass that much any more.
Happy gardening in Hawaii. Gardens are where people grow.