Loosing pressure is perfectly normal, and no it is not cheap Asian rubber that is the cause. If you want to see a tube that has to be inflated daily, go get a Michelin or Vittoria (European) latex tube. They are also more expensive and not as available as butyl rubber tubes. You will not find them at Wally's. But they result in a wheel with lower rolling resistance, so there are some people who prefer them.
Thicker tubes might loose less pressure, but they would add rolling resistance. If you rode back to back you could probably tell the difference, and you would prefer the thinner tubes. In the past, some people sold thicker "thorn proof" (they were not) tubes. Today "Slime" sells a very thin tube with a sealant in it. Both of these options will probably loose pressure more slowly than a conventional tube, but it is still a good idea to check the inflation regularly.
I have made it a point to check the pressure in my tires a couple times a week. I run fairly high pressure in my road bike, like 105 PSI in a 700CX25mm tire. Last year I put "Slime" tubes in my tires and they don't loose air as quickly, but I still checked them. This year I have switched over to tubeless. I checked them this afternoon and they were down to 80 PSI in about a week.
My mountain bike / winter commuter doesn't loose pressure as quickly, but it does loose pressure. The tires on it run at 30 PSI with Schwalbe butyl tubes in them for the winter tires and tubeless in the summer. Since the MTB doesn't go out as much in the summer, I check the pressure before each time I ride. On the tandem I run "Slime" tubes at 90 PSI in a 700C X 38mm tire. At home I check it a couple times a week. I carry a pump on tours and also check the inflation a couple times a week.
As far as I can remember, I have been checking inflation pressure in bicycle tires since the early 1980's. Compared to an automobile tire, they are very thin and the pressure is much higher.