host/tests: Phase 1 — picker load/filter/paging/error-retry as SX engine tests

Port the rest of the relate-picker's interactive behaviours from Playwright into
the SX harness, driving the real engine against the mock DOM:
 - load: the form's "load" trigger populates the results on first render
 - filter: a debounced "input" re-fetches and narrows the candidates
 - paging: revealing the load-more sentinel pages in the next page (outerHTML
   swap replaces the sentinel)
 - error-retry: a dropped fetch marks .sx-error, and the next request clears it

Models two browser natives the OCaml runner lacks: observe-intersection (a
recording stub the test fires to simulate the sentinel scrolling into view) and
the synchronous-timer retry (stripped in the error test — backoff timing is a
test-engine.sx concern; here we assert the visible state).

Mock-DOM completeness (run_tests.ml): firstChild/lastChild on elements, so
children-to-fragment can drain a parsed fragment into an innerHTML/outerHTML swap
target. (Also repairs one pre-existing web test that needed firstChild.)

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

hosts/ocaml/bin/run_tests.ml

@@ -2812,10 +2812,13 @@ let run_spec_tests env test_files =
               | "insertAdjacentHTML" | "prepend" | "showModal" | "show" | "close"
               | "getBoundingClientRect" | "getAnimations" | "scrollIntoView"
               | "scrollTo" | "scroll" | "reset" -> Bool true
-              | "firstElementChild" ->
+              | "firstElementChild" | "firstChild" ->
+                (* the mock treats element children and child nodes alike, so
+                   firstChild == firstElementChild — children-to-fragment walks
+                   firstChild to drain a parsed fragment into a swap target. *)
                 let kids = match Hashtbl.find_opt d "children" with Some (List l) -> l | _ -> [] in
                 (match kids with c :: _ -> c | [] -> Nil)
-              | "lastElementChild" ->
+              | "lastElementChild" | "lastChild" ->
                 let kids = match Hashtbl.find_opt d "children" with Some (List l) -> l | _ -> [] in
                 (match List.rev kids with c :: _ -> c | [] -> Nil)
               | "nextElementSibling" | "nextSibling" ->