Zack, I think, for many people, bonding with Hysol does result in a somewhat stiff seam, but it doesn't have to. If you want to improve your chances of achieving a very flat and flexible, nearly invisible seam, I would recommend three things:
1. Use Hysol U-09LV (the low-viscosity version, available from Fastenal), not Hysol U-09FL, and store the glue in the fridge. Since switching to the LV I have not gone back to the FL. The lower viscosity allows you to get a much thinner film of glue between the surfaces, and this gives a much flatter, more flexible finished seam. It also increases the apparent open time a bit because it takes longer to get unmanageably gummy. Storage in the fridge also increases the open time and improves the shelf life.
2. Use very, very little glue. A very small, threadlike bead of glue is plenty for a 1/2" to 1" wide simple overlap seam if you clamp it tightly. You can always go back and add a little more at the edge if you added too little, but you can't remove any if you used too much. Don't bother spreading the glue at all. just put the bead right down the middle of the surface to be glued, and press the two surfaces together.
3. Clamp! Clamp it very tightly. In my opinion, this is one of the most common mistakes that people seem to make with Hysol. There is no risk that you will force all the glue out of the seam. Use two boards at least the length of your seam (I use smooth, straight hardwood moulding). They should be at least two inches wide, to avoid a ridge of squeezed-out glue at the edge. Cut strips of LDPE painter's plastic and put them on top of and beneath the glued seam, between the seam and the wood. The plastic will be easy to peel away the next day. Buy a dozen or more (I sometimes use two dozen) 2" C-clamps from Harbor Freight or some other discount tool shop. Use these to clamp the boards together very tightly. Put the clamps as close together as you can. Leave the clamped seam in a warm place overnight. The bond will take about 48 hours to achieve full strength.
If you do these three things, you should be able to obtain an almost imperceptibly thin seam that hardly differs in flexibility from the surrounding material. If you don't clamp, or you add an excess of glue, you'll get a stiff lump, in my experience. I hope this helps.