I agree, this is the way to do it. One economical alternative is PL
Premium polyurethane construction adhesive. It comes in a cylinder for
a caulk gun. It's less expensive than most poly glues, and it works
wonderfully. It comes out about the consistency of peanut butter (same
color too). Like the other polys, it foams while curing and expands.

The only problem is that if you don't use it all in one application (I
never do), it cures inside the tip and seals shut. This is easily fixed
by drilling out the cured stuff when you need to use it.

This is VERY strong glue. I originally started using it to splice
together 2-liter bottles to make water rockets. A 4-inch diameter seam
could take 100 psi with no problem. That comes to 1256 lbs. of force
held together by a 12.5 inch-long smear of the stuff. And it was
getting that kind of adhesion on plastic!

Yeah, it'll hold a chair together.

Don't use water to clean it up. Just dry shop towels. If you get it on
your hands (don't), use dry rags to get it off. If you use water, it
will cure really fast and set up. You may even want to wet surfaces
before application, but I have no evidence that this helps. I've never
had a PL Premium joint fail at the glue, applied wet or dry.

-Mike