We do like to debate details, dont we? Even the predictable “Just train your horse!” response. :lol:
So much depends upon the situation and the horse. In this case, the horse belongs to the OP, so she can require the way she prefers. Period.
In the more general discussion, many of us take shortcuts in some situations. I can see leading with the reins over the neck when you are going from the tack up area in your barn to the mounting block in your attached arena with your generally calm horse. I might do that except I tend to break another rule and often dont put the reins over the neck before bridling so I have the reins down to lead with anyway.
For walking any distance, having the reins down just allows me to lead more correctly with the horse respecting my personal space. I also have a much better chance of maintaining control in the event of an unexpected spook. For me, that is a more important consideration than having the reins flopping should the horse get loose.
And if there is a need to walk more with the bridle on, the reins are secured and there is either a halter or at least a lead attached to the bit adapter.