Sidewalk Repairs

Brownstone Agency had requested that I report minor damage to my sidewalk to 311–essentially to “drop a dime” on myself. I just did that and got a service request number. The 311 operator said that I’ll get a violation from DOT in 30 days. I’m aware that I’m responsible for repairs to my sidewalk (unless they’re caused by tree roots) although the City is liable for any injuries caused by the sidewalk condition in front of my one family house. It used to be the case that if I do nothing after receiving the violation the City, in the fullness of time, will send a contractor to do the repair and bill me. This used to be the simplest and most economical solution; is that still the case?
FWIW I reported damage to my sidewalk three years ago (the last time my BA insurance was renewed) and NOTHING ever happened. That’s OK with me because the sidewalk damage is very small. OTOH I wouldn’t mind paying for the repair.