You see... My issue with using the recharge mechanic listed above is that it ends up overpowering the character far too much when compared to other characters in 5e unless you Nerf even weapon damage for a character that uses them.

Your method, by your own admission, allows a super attack every round for the first 5-6 rounds, and another every other round in every fight. That is more powerful than even 4e characters are.

The best way to emulate 4e characters is to shorten the length of short and long rests to 4e standards. At that point, the system functions very similarly to 4e with minor alterations, the 4e style characters still have the same attack rate, and everyone is relatively balanced.

Only basic attacks are really an issue, and that is simply weighing the balance between one attack that does insane damage (which is the forte of rogueish characters) or multiple attacks (more fighter-esque characters).