What should we be feeling with L-theanine?

I never found it affected me.

With good green tea that has lots of it (shaded tea, but doesn't need to be gyokuro, Sencha Momiji is my favorite, cold brewed over night, then strained in a french press), I usually feel a slow but steady rise in wakefulness and around the time where I would start to get jittery (say, like when drinking a few too many cokes), it transitions over to an incredible calm mixed with a feeling of... wideness. Like your FOV is actually expanded by a few degrees, on many levels. Close your eyes and it feels like being at peace in a kind of Zen state. But... with a little buzz, very much an eager, productive peace where things line up and you can approach them.

It's probably different for different people, but I can highly recommend trying to "get some good stuff" to see whether it can do the same for you. I wouldn't want to miss it.

This!

Another curious thing I have noted: While living in Japan I was binging on houjicha[0] (which has significantly less caffeine due to the roasting process) on late evenings so I would still be able to sleep. After several strong pots (we are talking litres of 1:1 water/leaf ration) I would often get into a different mind-space that is pretty close to what you are talking about. I tried to research if other catechins (like l-theanine) also disappear in the roasting process or if they could be the explanation, but weren't able to find anything to back this up.

0: https://en.wikipedia.org/wiki/H%C5%8Djicha

I haven't really taken it as a supplement/extract by itself, but drinking lots of strong green tea gives me distinct but different effect from coffee/energy drinks/caffeine pills.

It mostly seems to negate the negative effects of caffeine (apart from problems falling asleep) and gives a mood-boost and counteracting stress. It only goes so far of course, over-consumption of tea will still yield anxiety and dependence, but the threshold for this is a lot higher.