Charging 3.7 V Li Ion battery with 5 V DC
A Lithium battery uses Lithium ions.
A Lithium battery that is discharged to a voltage too low creates solid Lithium that is very unstable and flammable. 
A Lithium battery charger circuit is supposed to detect a low battery voltage then attempt charging with a very low current. If the voltage does not rise then the charging is stopped and an error is shown.

Your simple battery charger will probably cause the battery with a voltage that is too low to explode and catch on fire.

A Lithium battery charger circuit powered from only 5V probably uses a low-dropout voltage regulator circuit.

A Lithium battery charger circuit senses that the charging current has dropped on a battery at full charge then disconnects the charging since trickle-charging is bad.

Your simple battery charger will probably overcharge the battery continuously which is bad.

A simulator circuit will not smoke, explode or catch on fire but your simple circuit will when actually built.