It looks like the 3000T will just notify you if you try to pull too much current from the probe port, is that correct?
Yes, correct. You'll get an overload indicator and the scope will tell you to remove the probes and reboot. The rule of thumb is that you can use two fully-loaded active probes simultaneously with the 3000T's interface. You can often get away with more depending on what you're doing.
So it would be pretty hard for a bad probe to kill the scope?
I will cautiously say "yes." The protection is generally pretty good, but a broken probe could do some damage (especially to the 50ohm path) depending on the type of probe failure.
In any case I think I might try powering up the probes before connecting them to the scope, and checking voltages and current draw.
For the 3000/5000/6000/7000 Series oscilloscopes, you get at most +-400 mA max current supplied by the scope for all the probes combined. We call it "Autoprobe lite"
For the 4000X/6000X/9000/90000A/S-Series (BNC connection) you get +-200 mA
per channel. We call it "Autoprobe I"
For the 90000X/Q/Z (SMA connection) you also get +-200 mA per channel. We call it "Autoprobe II"
Have fun!