Can I program a factory fresh Atmega328P through UART pins without burning a bootloader?
No, the ATmega's ISP does not operate over the UART pins but only the ISP pins.

To program it via a UART, you need to first load a bootloader, unless your chip was pre-programmed by the vendor, something some offer as a value-added service (or even as an essential stock item, with an Arduino type bootloader)