Are you sure you're getting 1 to 1.5 inches? How did you measure that?

If so then you are correct in your watering. 3x per week is a recipe for a dead lawn in many cases.

The reason the aerator only went 1 inch deep is the soil is hard. Try this: apply baby shampoo at a rate of 3 ounces per 1,000 square feet followed by a full inch of water. If you get any runoff before an inch is applied, then stop immediately, let the moisture sink in for 15 to 30 minutes, then continue watering until you get the full inch. A week later water the full inch again. A week after that, repeat the shampoo and irrigation. That should set your soil up to allow much deeper water penetration. You can test the hardness before and after by sticking a screwdriver in as far as you can.

Normally after the shampoo treatment, people abandon the idea of aeration. Since you have already aerated, make the most of it by trying to fill your aeration holes with water. Try to flood the soil and get water to penetrate deeper.

The idea of getting the water to penetrate deeper is to improve the beneficial microbe population in the soil. Normally there are fungi that soften the soil for you. When the soil has deeper moisture, the temperature and moisture levels are moderated which allows the fungi to thrive. That process takes a few weeks to happen, but once it does, it will last for years.

The way I tested the amount was by using the tuna can method. The cans are around 1.25"-1.5" deep. By the time I turn the water off they are full. I will be watering the lawn on Monday so I will do the scewdriver test then. How long do I wait after watering to use the screwdriver?

Try the screwdriver right before you water and right after to gauge the difference. It should be softer right away and remain that way for a day or so. To get the full effect of the shampoo, you should try the screwdriver again in a month after you water.

Well I just watered the lawn and for the most part the 6" screwdriver went in all the way very easily. But not surprisingly the brown areas, the screwdriver only went in about halfway and then hit hard soil.

That is good news. That means you have normal soil except for the part that isn't normal. I would apply the shampoo to the entire lawn but linger a little longer on the brown areas. Applying more, even tons more, will not hurt anything.

Yeah I'll do the shampoo treatment on the entire lawn and I'll double the amount for the back lawn. Last night I watered the back lawn and basically the screw driver went in maybe a inch in most of the lawn. There were very few areas that the screw driver went in 6 inches. But that does not surprise me since the back lawn is used all the time.