You're talking about leaving one side of the girth attached to the saddle more or less more or less permanently?
I always take mine off. I like to tighten one hole at a time, alternating sides, when I tack up, so I may as well start with the girth completely off. I also run the girth through the loops of the saddle pad, so if I didn't take the girth off completely I'd have to put the saddle, pad, and girth on all at the same time and I don't believe I'd get the pad on without a bunch of wrinkles that way. I do have a friend who leaves her saddle, pad, and girth all connected between rides and it doesn't look like any less effort once she gets everything all straightened out.
How does your horse express that he prefers the girth to remain attached on one side?
“The horse is a mirror to your soul. Sometimes you might not like what you see. Sometimes you will.” - Buck Brannaman
"Nothing forced can ever be beautiful." - Xenophon