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/root.ts

@@ -4,6 +4,7 @@ import { pathToFileURL, fileURLToPath } from "node:url";
 import { servers, globalRootMarkers } from "../server.ts";
 import type { ServerConfig } from "./types.ts";
 import { UnsupportedExtensionError } from "./types.ts";
+import { isOnPath } from "./util.ts";
 
 // Resolve File URI To Local Path
 export function uriToPath(uri: string): string {
@@ -15,11 +16,24 @@ export function pathToUri(p: string): string {
   return pathToFileURL(path.resolve(p)).toString();
 }
 
+// Server Availability Cache - Checked once per process lifetime per server.
+// Avoids repeated filesystem lookups on every tool call.
+const serverAvailability = new Map<string, boolean>();
+
+// Is Server Available - Returns true if the server binary is on PATH.
+// Result is cached for the lifetime of this process.
+export function isServerAvailable(server: ServerConfig): boolean {
+  if (serverAvailability.has(server.id)) return serverAvailability.get(server.id)!;
+  const available = isOnPath(server.command, process.env);
+  serverAvailability.set(server.id, available);
+  return available;
+}
+
 // Pick Server By File Extension - match[] entries are matched against the
-// file's extension (no dot). First server in the registry wins.
+// file's extension (no dot). First available server in the registry wins.
 export function pickServer(filePath: string): ServerConfig {
   const ext = path.extname(filePath).replace(/^\./, "");
-  const hit = servers.find((s) => s.match.includes(ext));
+  const hit = servers.find((s) => s.match.includes(ext) && isServerAvailable(s));
   if (!hit) {
     throw new UnsupportedExtensionError(`.${ext}`);
   }