refactor(root): extract isOnPath and add extension-side server qualification

Extract isOnPath() to shared src/util.ts so both the daemon (client.ts)
and extension (root.ts) can use it. Add isServerAvailable() with a
per-process cache to pickServer(), skipping servers whose binary isn't
on PATH before sending requests to the daemon.

This avoids wasted daemon round-trips for missing binaries and sets up
for upcoming multi-server diagnostics fan-out.

src/client.ts

@@ -10,35 +10,10 @@ import type {
   InitializeParams,
   PublishDiagnosticsParams,
 } from "vscode-languageserver-protocol";
-import * as path from "node:path";
 import type { ServerConfig } from "./types.ts";
 import { ServerNotFoundError } from "./types.ts";
 import { findRoot, pathToUri, uriToPath } from "./root.ts";
-
-// Is On PATH - Returns true if `cmd` resolves to an executable via the
-// supplied PATH. Absolute/relative paths are checked directly.
-function isOnPath(cmd: string, env: NodeJS.ProcessEnv): boolean {
-  const isExec = (p: string) => {
-    try {
-      fs.accessSync(p, fs.constants.X_OK);
-      return true;
-    } catch {
-      return false;
-    }
-  };
-  if (cmd.includes(path.sep)) return isExec(cmd);
-  const exts =
-    process.platform === "win32"
-      ? (env.PATHEXT ?? ".EXE;.CMD;.BAT").split(";")
-      : [""];
-  for (const dir of (env.PATH ?? "").split(path.delimiter)) {
-    if (!dir) continue;
-    for (const ext of exts) {
-      if (isExec(path.join(dir, cmd + ext))) return true;
-    }
-  }
-  return false;
-}
+import { isOnPath } from "./util.ts";
 
 // LspClient - Thin wrapper that spawns a language server, performs the
 // initialize handshake, auto-opens a file, and exposes sendRequest so the