Definitely the bird's wanting to be with you is pretty much directly proportional to the relationship you have built up with him, just like with most other pets. So it is something that builds up over time, and of course some pets are more shy than others so that factors into it. But I'm sure after a few more months of daily playing that your keet will get more and more 'family oriented' and want to be out with you guys (i.e. his flock) vs alone in his cage. Parakeets are VERY flock oriented and don't like to be alone!
It is sometimes tricky to help a bird get over the fear of the bathtub, but really, they are naturally drawn to water in general and love to be clean. So keep trying daily with the water. Have the water lukewarm (not cold, not hot) so it's comfy on the skin, and wiggle your fingers in it to make it splash a little. They love the sound of running water. I have a page on the site about helping to train your bird to get into the bathtub, it just takes putting the keet on the tub each day until he gets the idea
Try putting one of the gyms right next to the cage and things right on top of the cage. That way he gets used to the idea of playing "on" the cage - just on the other side of it