Here are my 3 concepts. I noticed that you want to be able to easily transfer to labels, packaging & T-shirts. I worked for a promotional company (screenprinting and embroidery also) and having gradients in your logo is not going to work on some items. Also will cost a lot if you have more than 1 or 2 colors, just for your FYI. A lot of our vendors cant print gradients either.
Yes, in screenprinting it doesnt look good and in embroidery its hard to do too and doesnt look right. Though you could always pick a logo and have 2 versions of it, one with the gradient and one without (solid colors)