I'm going through the same thing as you.
First, you have to determine if your house was pre-wired for a security system. Yes, i know you had one, but the question is, did Brinks do the actual wiring, or did the builder do the wiring, and Brinks used it? My house WAS prewired, meaning all the doors and windows were wired, but the ADT system added just used some of these wires.
In any case, it best to get rid of the panel and consoles, but use the wires that are there. if you need widows and doors NOT wired, its a million times easier to add wireless sensors. In fact, you could go all wireless, but its generally a bit more expensive for wireless than wired, but labor is much less.
ELK and HAI/Leviton are the two main home controllers that are alarms as well. The Admco Vista20P are good basic alarm only systems. For me, if you ar5e doing all this work, I'd spend a bit more for a full home automation capable system while your at it, but it is up to you.
O.K. back to the wires. Yes, the wires will work, as will the door and window sensors, BUT there is this thing called the End-of-Line (EOL) resistor that varies in value from one system to another. The CORRECT way to install these resistors is at the sensor itself, but almost none of the cheap systems like Brinks do that. Instead they are sometimes installed in the panel. If that is the case, they are easy to replace. If they are installed at the sensor, they are harder to replace, but their are workarounds.
So yes, using the existing wires is possible, but even better is if your home was pre-wired from the start.