Yes, you could, but you are missing the whole point of the video. You don't need, and should not use, resistors to set the current passed through one, two, three... or however many... LEDs!
That you used resistors in the beginning was so you could learn about how LEDs work, how much voltage they require to "light up" and how to use resistors in simple way to set the current drawn by the LED or LEDs (if more than one). All that is still valid and was a necessary part of your learning experience with LEDs, but it is time to move on to more efficient ways. Along the way, don't forget to visit LEDs of different colors, which all have different operating voltages that depend on their color. Red (and infra-red) is the lowest voltage and the voltage increases as the color moves though yellow, orange, green, blue, and ultra-violet. So-called white LEDs are actually ultra-violet LEDs illuminating a phosphorescent light emitter, reminiscent of fluorescent lighting.
Unlike your 1.5 V incandescent lamp, which requires a fixed (constant) voltage to operate properly, your LED requires a fixed (constant) current. Power, as received at the convenience outlets in your house or as delivered from a battery, is provided at a fixed (constant) voltage. It requires a bit of circuitry and solid-state (semiconductor) components to efficiently deliver power to your LEDs at a fixed (constant) current. Thankfully, it is now quite inexpensive as well as efficient to do just that!
You will notice that the circuit board in the video you posted has a LOT of probably unfamiliar components. Never mind that! The engineering and product testing has already been done for you. Just purchase the module and connect four wires (plug and chug) for around a buck American or thereabouts. Voila! You will now have an adjustable, constant-current, power source for all your LED experiments. As a bonus, most of these modules can also be adjusted to operate as adjustable constant-voltage power sources, which you will eventually need for experiments as you continue your journey into the fascinating world of hobby electronics.
Others here have mentioned the impracticality of using tiny 9V "smoke alarm" batteries for your experiments. Batteries, even if you use rechargeable ones, are expensive and have limited power availability and energy storage capability. However, to gain better operating life, you should purchase a battery cell holder for six size-D dry cells and wire the cells in series. Most battery holders are already wired with the cells in series, but check to make sure. This arrangement will provide an initial output of 9 VDC with at least one ampere of current capability. It should be more than enough until you venture into the area of power electronics with motors and high-wattage audio amplifiers.
For serious experimentation I cannot over-emphasize the need to employ the full faith and credit of your local electric utility to power up a "wall wart" capable of delivering at least nine volts at a few hundred milliamperes of current. Avoid so-called 5V USB wall wart cell-phone chargers because some of them can be quite dangerous. A real wall wart power supply has a substantial "heft" to it because it incorporates a transformer to isolate the power supply output from the power line of your public utility. This transformer may not necessarily be used at 60/50 Hz power line frequencies, but may instead operate at a much higher frequency as the power line is rectified to DC, the DC used to power an oscillator, and the oscillator used (with a high-frequency transformer) to deliver isolated power to the wall wart output. Again, all at a very inexpensive cost and high efficiency.
