HC-05 Bluetooth Module Blinking Red Light-Arduino Uno
There are several issues to consider:
• Program downloading
• HC-05 serial communication
• Bluetooth pairing

For the first issue:  The serial-over-USB connection between the development computer and the Uno ties up the Uno's hardware serial RX and TX pins.  There will be a conflict if the development computer transmits when the HC-05's TX is connected to the Uno's RX.  This can be resolved by disconnecting the HC-05 TX when downloading a program.

For the second issue:  The Uno schematic shows 1KΩ resistors between TX and RX on the ATmega328P and RX and TX on the USB chip (ie on an ATmega16U2, or a CH340, etc).  If the HC-05's TX drive is strong enough, it will overcome the drive from the USB chip and work ok.  Note, I would leave out the voltage divider (on the HC-05 RX line) that the video shows.

For the third issue:  The HC-05 in the video mentioned in the question has a blue LED that blinks steadily when the HC-05 isn't paired, and blinks at a long interval when it is paired.  Many HC-05's have red LEDs instead of blue; the LED color is not an issue.  Some HC-05's don't blink at all when paired; that's not an issue either.  What is an issue is that (according to description in question) your HC-05 keeps on blinking, so apparently isn't paired up.

Whether the HC-05 pairs with an Android app is more or less independent of whether you have the HC-05 connected to an Arduino; you should be able to pair your phone with the HC-05 when just Vcc and ground are connected to the HC-05.  If you are having trouble with pairing, try another app; try powering the HC-05 down and up; and try a different HC-05.  I mention trying another app because one serial-terminal app I installed on my Android phone needed the app or the HC-05 to be restarted to pair up after loss of pairing, and the serial-terminal app I replaced it with did not need those actions.  Note, pairing takes a few steps on my phone:  in Settings / Wireless, search for Bluetooth devices; select device; enter its passcode; run an app and tell the app which Bluetooth device to use.