Will my 12V power supply ever burn these IR leds blinking at 38kHz even without a resistor?
These LEDs are specialized on short pulses of high current, so pretty much what you're doing. The datasheet suggests that 100µs pulses can be driven by currents as high as 1A (pulse peak, see "basic characteristics" -> "Radiant Intensity"). 

With 38kHz your pulses are on the order of 15µs only, so you might be getting away with even higher currents. But note that the datasheet doesn't give any info above 1A (see figure 3, which might be of interest to you).

Your way of reaching those currents is dangerous for the life of the LED though. Don't drive them with 12V without a current limiting resistor, even if the resistor limits the current to something high like 1A. That's just how LEDs work, their differential resistance isn't linear. What's probably saving you right now is the batteries' inability to deliver such high currents in so short pulses. You're effectively not driving them with 12V, because your batteries' voltage drops temporarily. That's not so good for the batteries either, depending on their chemistry, or anything connected to the batteries (might brown-out). You could add a fat 47µF ceramic capacitor to buffer the energy in front of your output stage. That might help burn those LEDs :).

The reason you are getting worse results than the controller might also have to do with the angular distribution of the emitted light. The LEDs you're using are pretty focussed (see fig. 9), their half-intensity angle is just plus/minus 10°. Maybe you'd be better off with an LED that has more spread?