Actually, is a resistor really needed?
Without a resistor there, you need to get R1 just right. Too low and Q1 will overheat; too high and Q1 will not supply enough base current to Q2. The problem is, the ideal value of R1 (in the absence of a resistor in series with the collector of Q1) depends on the hfe of Q1, which can vary widely between different transistors having the same part number.
You say it's to "limit the collector current and power dissipation of Q1", but the collector current is coming down through R2, not coming out of the base of Q2.
No, it's mostly coming out of the base of Q2.
Q1 is acting to pull the base of Q2 low - a resistor in there will form a voltage divider with R2 and not pull the base down enough.
Only if the ratio of R3 (the new resistor) to R2 is too high.
The current through Q1 is controlled by the base current, which is limited by R1.
True; but the collector current of Q1 is the base current times the hfe, and the hfe can vary widely between different transistors of the same type.
The current through Q2 is whatever L1 requires.
current through Q2 is whatever L1 requires, provided that Q2 has sufficient base current.
I don't see what benefit an extra resistor would have, and I can see how it would stop it working.
The purpose of the resistor R3 is to ensure that Q2 gets enough base current to saturate, but not so much that Q1 overheats. Including R3 will only stop the circuit working if the ratio R3:R2 is too high.
You should choose R3 so that the base current into Q2 is around 1/10 to 1/20 of the current through L1 (hence R3 is 10 to 20 times the resistance of L1, because the voltages across L1 and R3 are very similar, and most of the current through R3 comes from the base of Q2). Then choose R1 so that the base current is at least about 1/50 of the chosen Q2 base current (I am assuming that Q1 has an hfe of at least 50). I would choose R2 to be about the same as R3.