It all depends on you want, and what you are talking or thinking about, when you ask What is an Arduino. There are two main things that I associate with Arduino: A family of Development Boards (maybe a hardware family of development boards) and a Software built to work with that development boards. This let me to answer you using that two focuses:
- The hardware focus
I understand that the Arduino boards were intended to bring the electronics and programming, closer to artists. Also, as any development board, the Arduino boards allows the user to test the most quantity of features of the core microcontroller used in the board (for Arduino UNO, the Atmega328P). Thus implying that the most of components included are to allow you a "safe" development workaround.
Now, when you have worked in a project so many time that you are ready to replicate your design, maybe for a commercial implementation, an Arduino board can take up more space that if you design your own board, well by you are not going to use all the pins or well by that you noted that you are not going to mess up the microcontroller or other components (maybe you did it the necessary times to know it) and you want to choose a smaller package that don't needs an easy microcontroller replacement, or finally because you want to include in a single board all the components and circuitry you used for your project and remove the ones you are not going to use, that is the case you mention, and you can do more things... if you want to work from a CR2032 battery you won't need the regulator and so...
- The software focus
When you decide to use your own circuitry, maybe you also want to program the IC by yourself, not through the 2K bootloader preprogrammed, but by your own ISP programmer. There are many workarounds using the Arduino IDE, but as apparently you don't want to use the Arduino IDE, the ISP programmer would be your programming interface.
A conclussion?: The minimal things you have to bear in mind to put in run the Atmega328P (and maybe most of microcontrollers) are 3: polarization, clock and Reset. The other things (the LC network you mentioned, the RC network I will mention, and so...) can improve the microcontroller stability and performance.
- Polarization. Depending on the board, the DC supply used can be either 5V or 3.3V. Check the voltage used by your board.
- Clock/Oscillator. See and reply the oscillator configuration used in the board. Then search where you can set up (in your IDE or code) the oscillator speed. Overall if you decide or want to use the internal oscillator, obtaining a minimal circuit or by playing with arduino config files. Note that, as example with the 328P, using the internal oscillator means that you are not going to reach the max number of MIPS the 328P can reach (with the internal you have 8MIPS, but you can use an external crystal up to 20 MHz to get 20 MIPS)... It really matters if you are going to do many processing tasks.
- Reset. Please do not let it open, send it to VCC through a 10Kohm resistor, and (if available) additionally to GND with a 100nF ceramic capacitor.