I do not know of any commercially available products alternative to velcro, but I'm a DIY-er, And have successfully mounted some accessories to the frame and handlebars using hose clamps.
Their advantage over velcro straps is that they can hold more weight and can better stabilize the load. A strap of old inner tube is enough to protect the paint. Since the clamp does not slide, it does not rub against the paint. The clamps are ugly, but allow for semi-temporary solutions.
Another option can be specially made plastic clamps (Similar to the ones used to mount reflectors and accessories). Maybe 3D printed. They would also be "visible" but could be designed to be more eye pleasing and to clamp on areas that do not obscure the logos.
Another option is to fabricate a bag that can be mounted to the water bottle cage eyelets. Some frames have two sets: on the upper side of the down tube and on the front of the seat tube. The bag would need an inner frame to be bolted to these eyelets. A simple aluminum flat bar should be enough. If only one pair of eyelets is available, only a smaller bag should be used as to not overload the bolting points.
For those who order a custom frame, this could be a serious option (ordering specially placed eyelets).
For those willing to modify a frame, eyelets can be installed appropriately. (Depending on frame design and material).
Edit / afterthought: Another way could be to build an inner structure for the bag that can be "press fit" into the bike frame or may have some bolts that make it "expand" so it tightens inside the front triangle.
The frame would need to have the inner shape of the front triangle and half-pipe (-ish) profile to avoid sliding out, at least on contact points. On the outside, the contact points should have rubber or similar material to avoid sliding.