I have a new shallow bathroom sink that has no overflow. I put in a bar-sink drain cover (to prevent anyone from closing the drain), and the water doesn't flow down the drain. It just stands there, until I run my finger back and forth over the holes in the drain cover. There is a vent pipe on the main drain pipe. Is there an easy solution to this? Would a drain pipe vent solve the problem?
If by "bar-sink drain cover" you mean a perforated strainer like this:
then it might be related to the relative "softness" or "hardness" of your water supply. Where I am from we have "hard" water (high in mineral content) which has high surface tension and will not drain as easily through fine strainers. "Soft" water has less or no dissolved minerals but has dissolved sodium, and lower surface tension. Get a gallon of distilled water at the market, or mix 1/2 gallon of water with a 1/4 cup of table salt and see how that drains. If it drains fast, then you probably have to buy a strainer with bigger holes or enlarge the holes in your strainer. Or buy a water softener. Or best of all, install a sink with overflow protection.