Shortest by what measure?
By angular change. When pilots are expected to turn onto a new track or heading they are expected to turn "the shortest way" which means through the angle that is the smaller of the two options available.
Also, by my reckoning, if a medium jet turned left out of the hold it would intercept the initial just in time to start the descent. If it turned right it would have a nm to spare.
When cleared to leave the hold the aircraft must route to the IAF. From that moment on the hold is no longer an issue. The IAF in this case is a fix where a turn is specified and when the aircraft reaches ther IAF it is expected to make the turn towards the outbound track. This turn has nothing to do with the hold and the turn should be made towards the outbound track no matter what direction the turn fix is approached from.
Neither tailwinds or DR is relevant as a right turn would stay with the holding pattern until a track of 300º had been achieved to intercept the initial.
It is very relevant.
The only time it is not relevant is when the aircraft is tracking inbound to or outbound from TGO. While turning the aircraft is being drifted downwind. From the completion of the turn until the aircraft has established on the outbound radial the aircraft is operating on DR. The amount the aircraft drifts during the turn depends on how long the aircraft spends turning. the amount the aircraft drifts on the DR leg to intercept depends on what wind info is available and how the pilot uses it.
The easiest way to look at this is to remember that you want to spend the minimum amount of time with no tracking information. You do not help yourself in this regard by increasing the time spent turning.
Finally, it is very possible that having turned the wrong way, when a track of 300 is acheived the aircraft will be positioned in such a place that a track of 300 will not intercept the outbound radial and will fail tom return to TGO. not a good place to be eh?
It doesn't matter what way you cut it, staying south of TGO for any manoeuvring is covered by the hold, the initial and the arrival buffer areas.
In the buffer area, you do not have obstacle clearance guaranteed. Within the hold area obstacle clearance is guaranteed. In the holding area the obstacle clearance is based on the aircraft following a specific procedure and always regaining the inbound track and following it to the hold fix - TGO.
You fail to recognise that with a strong northerly wind, your aircraft will establish on the track of 300 some way south of the still air point for the start of the hold outbound leg. The aircraft will then track 300 which will cause it to miss TGO and leave the holding area. How do you know that where the aircraft leaves the holding area it will be within the middle half of the Initial Approach area (wher obstacle clearance is assured? Or that it will be within the turn area designed for a correct direction turn?
The turn fix is TGO. Treat it as such - a turn fix - a turn at the IAF. Forget the hold as soon as you are cleared beyond the IAF and route to TGO where you make an appropriate turn to establish outbound and not to fly tyhe hold again (unless you want ATC to repeat the clearance to proceed beyond the IAF because it looks to them that you are flying another hold) and they do not expect you to turn the wrong way.