"Better" depends on your design criteria. Simple, inexpensive, and efficient are some you might consider. Form factor (how big it is and where will it fit) is usually a desirable consideration. A 47,000 μF capacitor is likely to take up more space than, say, a Microchip PIC which can perform the same timing function.
Using a transistor as a "variable resistor" to control the load current to a string of LEDs is pretty simple but not very efficient. What if someone asked you to apply your circuit principle for use in a large auditorium to slowly dim the "house lights" prior to the beginning of a performance? Sure, this is not the original purpose for your circuit, but why not think of other applications where a controlled dimming effect is desirable? A new circuit could be "better" in the sense that it is more versatile if the circuit can be applied to more than just dimming a few LEDs.
When controlling large amounts of power, a transistor operating in its linear mode is very inefficient. That does not mean you should not do it. The popular three-terminal series regulator uses a transistor in this manner. However, a circuit that uses a transistor in switch mode (off and saturated on) is more efficient, because the only time significant power is dissipated in the transistor is when it switches from "off" to "on" and from "on" to "off". This switching action can be extremely fast (microseconds) and the ratio of "on" time to "off" time (duty cycle) is easily controlled to deliver a variable amount of power to the load. This is called pulse-width modulation (PWM) of the power source.
A disadvantage of PWM to control power applied to a load is added complexity, but with modern integrated circuits complexity is a given that is built-in at no additional cost. Amazingly enough, complexity at the level of integrated circuits often costs less than a discrete analog component design, of which your circuit is an example. Does this mean one circuit approach is "better" than the other? Not necessarily. It all depends on your design objectives, your criteria.
For example, consider an animated advertising display operated by a small permanent-magnet (PM) DC motor: potential customer presses a button, or just walks by the display, triggering a PIR sensor, and the motor starts. The mechanical animation runs for awhile and then slows down to a stop, awaiting the next activation. It could have been designed to just abruptly stop, but market research has shown that slowing down the display is a more effective "attention getter" to passers by. (I am making this part up, but it sounds plausible, right?)
As you learn electronics you will accumulate a rich set of tools (previously designed circuits) that you will use in future designs. The circuit you have shown us is just one of those tools. There may be "better" tools to use for different or similar applications, and it will be your job to select the "best" tool for the job. I would compare this to my woodworking hobby, where I have many saws that bear this similarity: the all cut wood. The saw I choose to use depends on how I want to cut the wood. I would not use my portable contractor's saw to cut pins and tails for a precision Dovetail joint
, for example. And I would not use the saw that I use to build Dovetail joints to build a house.
If you want to "improve" your basic circuit consisting of a PIR sensor, a string of LEDs, and something in between to create the slow dimming action, I would investigate PWM controlled by a small PIC microprocessor. You can leave the 47,000 μF capacitor behind for use in filtering rectified power in another project.