Why is the voltage regulator's output current not what I'm expecting?
The 60 mA is under Absolute Maximum Ratings, which means you never should exceed that at the expense of damage to the regulator. Since you can get twice that out of it it appears that there is no internal current limiter. Which is BAD, every regulator nowadays has current limiting.  

It appears that you'll have to make sure yourself that the 60 mA is never exceeded. The datasheet's electrical characteristics and the graphs are specified at 20 mA.  

There are lots of regulators with better specs available, like uA ground current instead of mA. Also the look of the datasheet doesn't give me much confidence.  

edit
Possible alternative: Seiko S-812C30B:


output voltage 2.0 V to 6.0 V, selectable in 100 mV steps  
2% output voltage accuracy  
up to 16 V in  
low ground current of 1 &micro;A typical, 1.8 &micro;A maximum (that's 1/1000th of the LA5003!)  
LDO  
short-circuit protected  
power-down mode  
guaranteed stable without output capacitor  


I'm a fan. In the past I didn't recommend it, though, because of it's bad availability, unless you needed 100k/year. But now I see you can get them at Digikey too.