You cannot do this. Initiative is not a skill; Jack of All Trades just applies to all ability checks.
The section on "Skills" lists out all the skills:
- Sleight of Hand
- Animal Handling
Note that nowhere in this list is "initiative" listed as a skill. Instead, the "Initiative" section states:
When combat starts, every participant makes a Dexterity check to determine their place in the initiative order.
Rolling for initiative is a raw Dexterity check, with no associated skill. The Bard's Jack of All Trades feature states:
Starting at 2nd level, you can add half your proficiency bonus, rounded down, to any ability check you make that doesn’t already include your proficiency bonus.
This feature applies to any and all ability checks, not just ones that are associated with skills, which is why it can benefit initiative rolls despite them not being a "skill check".
This means that you could not select initiative as one of your proficiencies because initiative is not itself a skill.