Two days in Tbilisi may not be enough to see everything
It's almost enough! I lived there for almost a year and I quickly found the place tiresome. Most of the city is just the old district. It's a lovely area but it's a little thin on things to do for daily living. Some nice cafes and bars around but not a ton of events in the city. You actually visited about 90% of what the city offers in this post. Mtatsminda Park is lovely in the evenings, but the only other thing I'd suggest is on the other side of the city around Marjanishvili Station. I think it's a bit nicer, very old and beautiful architecture. The roads are smaller (also bumpy and cobblestone) and less noisy, and there's a beautiful garden tucked away that still has that early 1900s feeling to it. As well as Fabrika which is a more urban side to the city where a Soviet factory has been turned into a hostel and space for bars, cafes, and some other stuff.
Once you step out of Old Tbilisi and more north the city is just chaos. Riddled with crime and drug needles. Residential Soviet spaces that haven't been kept very well throughout the years.
RE: Arrival in Tbilisi: First Impressions