It is well known that the shifting/power transfer of the 6 speed transmission is dismal in an effort to enhance fuel economy at the expense of performance. Factory software updates have minimal improvement. I found the best way to improve city driving performance is to operate in tow/haul mode on the transmission. This will hold the lower gears longer and will down shift better instead of upshifting too soon and keeping the transmission in higher gears at lower speeds. Give it a roll and see if it works for you.
Yes it will hurt mileage but gives much better performance for city driving. I used to manual shift the same way you described but found that the tow/haul mode does the same without the bother of shifting.