Confirmed Bug: Why does Eliminate fail while Solve succeeds?

I'm not sure what the hang-up for Eliminate[] is. This is fast:

GroebnerBasis[{p1, p2 - r}, {c, r}, {z}]
(*
  {4096 + 8192 c^2 + 12288 c^3 + 12288 c^4 + 12288 c^5 + 4096 c^6 - 
    768 r - 256 c^2 r + 256 c^3 r + 256 c^4 r + 48 r^2 - 16 c^2 r^2 - r^3}
*)

I offer the following approach. Let

soln = Solve[{p1 == 0, p2 == r}, r, {z}]

Then find the polynomial that has these roots:

0 == Times @@ ((r - Last[#]) & /@ Flatten[soln]) // 
 Collect[#, r, Simplify] &
(* 0 == -4096 (1 + 2 c^2 + 3 c^3 + 3 c^4 + 3 c^5 + c^6) - 
  256 (-3 - c^2 + c^3 + c^4) r + 16 (-3 + c^2) r^2 + r^3 *)