One thing if your bike comes with tubeless, but to convert to them because of a flat is dumb.
I presume you have a road bike. Personally, I find my mountain bike with front shocks is way more practical/useful. I can ride both road and off-road trails. I haven't had a flat tire since I was a kid.
As for saving 200-300 grams (which equals a measly 200-300ml of water), unless you're racing that's not going to make a significant difference. Besides, most people could easily loose 10 times that weight with very little effort.
Sounds like a coincidence. If there's a leak in the tube, something caused the puncture. Either the rim, something in the tire or a result of puncturing it when reinstalling the tube/tire. Do you have two tubes with punctures? If so are they in the same location?
If the bike has been sitting for a while, it's likely developed a flat spot. If he tire is in otherwise good shape, inflating it to the maximum pressure may resolve or at least help the situation.