Yes, the Bard adds 1/2 proficiency to initiative.
In 5e all checks are ability checks. This is why every check in published materials is listed as Ability (Name), such as Strength (Athletics).
Initiative is a Dexterity check. Under Initiative in the PHB:
every participant makes a Dexterity check to determine their place in
the initiative order. (p. 189)
If you can add an ability modifier to a roll, it's an ability check (A saving throw or an attack is a separate kind of d20 roll). If you have a Skill related to that Ability, and you are proficient in that skill, you add your proficiency bonus.
Initiative is a Dexterity check with no other Skill attached to it. Things that modify Dexterity checks work. Other things modify your ability checks as well - such as adding all or half of your proficiency bonus. The Bard has one example of this.
The Champion Fighter's L7 feature has another:
Starting at 7th level, you can add half your proficiency bonus (round up) to any Strength, Dexterity, or Constitution check you make that doesn’t already use your proficiency bonus. (BD&D 26)
Since Initiative is an ability check that uses your Dexterity, the Champion Fighter also gets half their proficiency to added to their initiative roll.