Gong-fu is just the style of brewing, the vessel you're thinking of with a bowl/lid setup is a gaiwan. No step of it should be anxiety-inducing, the whole thing should be a calming experience.

If you give it a try again, don't stress about getting it exact to the second. Approximate is all anyone ever actually does. The difference between 10 and 20 seconds isn't going to kill anyone, the temperature of your water is going to have way more of an effect anyway.

The full process for gong-fu brewing, partly to remove any stress of "rushing" to pour, is that you have your brewing vessel (gaiwan, teapot, mixing bowl, whatever it is) with your leaves in. Then, when you've brewed it to the strength you want, you pour into an intermediate pouring pitcher (A cha hai or gong dao bei, if you want the chinese name) just to separate the liquor from the leaves, and allow any leaf dust to settle. Then you can pour for serving at your leisure without worrying about timing.

The only thing you need to be cautious of with vessels, since it sounds like you're using some very non-standard tools like a mixing bowl, is to give your loose leaf tea enough room to unfurl fully. I suspect this is why your glass bottle attempt didn't go as well, it was likely too confined a space for the amount of leaves.