Bad idea. Always use aluminum or brass to drive bearing races. Impact sockets are hardened and will damage your races.
EDIT: You don't care about what you're hitting the socket with; it's the socket hitting the race you care about.
A driver which has a diameter close to the race's is always preferable to a (soft) punch or drift. The bore is softer than the race and if you drive one side in too far, you'll damage the bore. That's why a round driver and a press or clamp is always preferable. It's not impossible to use a drift or (soft) punch, you just have to be really careful.
You can check your punch to see if it's hardened. If you can scratch it with a file, it's soft.
Impact sockets are less hard than regular sockets and certainly softer than a bearing race. They will work fine if you hold them against the race and tap the socket with a hammer.
You can use the old race just to get it started and a little ways in, so the old race does not get stuck in. Once the new race is started square, you can then use something brass to keep it moving, but there's little value to using the old race from that point - especially to shoving the old race in so far it's now also stuck in. Just be absolutely double sure you are putting the new race in with correct orientation. When using a flipped old race, it can cause you to not focus on that.
this.
use a zip cut blade and slice through it. Works an absolute treat. perfect size every time.
I've previously used brass drifts (still do), but using the old race is a revelation.