refactor(doctor): show detected LSP servers and extensions instead of hardcoded counts

Replace the hardcoded 4-server list in doctor LSP check with getAllServers()
from server-resolution.ts, which covers all 40+ builtin servers plus user
config. Output now shows server count with supported extensions, and verbose
mode expands to per-server detail lines.

Status:  LSP 3 servers (.go, .py, .pyi, .ts, .tsx)
Verbose: LSP 3 servers (.go, .py, .pyi, .ts, .tsx)
           typescript (.ts, .tsx, .js, .jsx)
           pyright (.py, .pyi)
           gopls (.go)

Closes #2587

src/cli/doctor/checks/tools-lsp.ts

@@ -1,25 +1,9 @@
-import type { LspServerInfo } from "../types"
-import { isServerInstalled } from "../../../tools/lsp/config"
+import { getAllServers } from "../../../tools/lsp/config"
 
-const DEFAULT_LSP_SERVERS: Array<{ id: string; binary: string; extensions: string[] }> = [
-  { id: "typescript-language-server", binary: "typescript-language-server", extensions: [".ts", ".tsx", ".js", ".jsx"] },
-  { id: "pyright", binary: "pyright-langserver", extensions: [".py"] },
-  { id: "rust-analyzer", binary: "rust-analyzer", extensions: [".rs"] },
-  { id: "gopls", binary: "gopls", extensions: [".go"] },
-]
+export function getInstalledLspServers(): Array<{ id: string; extensions: string[] }> {
+  const servers = getAllServers()
 
-export function getLspServersInfo(): LspServerInfo[] {
-  return DEFAULT_LSP_SERVERS.map((server) => ({
-    id: server.id,
-    installed: isServerInstalled([server.binary]),
-    extensions: server.extensions,
-    source: "builtin",
-  }))
-}
-
-export function getLspServerStats(servers: LspServerInfo[]): { installed: number; total: number } {
-  return {
-    installed: servers.filter((server) => server.installed).length,
-    total: servers.length,
-  }
+  return servers
+    .filter((s) => s.installed && !s.disabled)
+    .map((s) => ({ id: s.id, extensions: s.extensions }))
 }

src/cli/doctor/checks/tools.ts

@@ -1,6 +1,6 @@
 import { checkAstGrepCli, checkAstGrepNapi, checkCommentChecker } from "./dependencies"
 import { getGhCliInfo } from "./tools-gh"
-import { getLspServerStats, getLspServersInfo } from "./tools-lsp"
+import { getInstalledLspServers } from "./tools-lsp"
 import { getBuiltinMcpInfo, getUserMcpInfo } from "./tools-mcp"
 import { CHECK_IDS, CHECK_NAMES } from "../constants"
 import type { CheckResult, DoctorIssue, ToolsSummary } from "../types"
@@ -13,14 +13,12 @@ export async function gatherToolsSummary(): Promise<ToolsSummary> {
     getGhCliInfo(),
   ])
 
-  const lspServers = getLspServersInfo()
-  const lspStats = getLspServerStats(lspServers)
+  const lspServers = getInstalledLspServers()
   const builtinMcp = getBuiltinMcpInfo()
   const userMcp = getUserMcpInfo()
 
   return {
-    lspInstalled: lspStats.installed,
-    lspTotal: lspStats.total,
+    lspServers,
     astGrepCli: astGrepCliInfo.installed,
     astGrepNapi: astGrepNapiInfo.installed,
     commentChecker: commentCheckerInfo.installed,
@@ -57,7 +55,7 @@ function buildToolIssues(summary: ToolsSummary): DoctorIssue[] {
     })
   }
 
-  if (summary.lspInstalled === 0) {
+  if (summary.lspServers.length === 0) {
     issues.push({
       title: "No LSP servers detected",
       description: "LSP-dependent tools will be limited until at least one server is installed.",
@@ -109,7 +107,7 @@ export async function checkTools(): Promise<CheckResult> {
     details: [
       `AST-Grep: cli=${summary.astGrepCli ? "yes" : "no"}, napi=${summary.astGrepNapi ? "yes" : "no"}`,
       `Comment checker: ${summary.commentChecker ? "yes" : "no"}`,
-      `LSP: ${summary.lspInstalled}/${summary.lspTotal}`,
+      `LSP: ${summary.lspServers.length > 0 ? `${summary.lspServers.length} server(s)` : "none"}`,
       `GH CLI: ${summary.ghCli.installed ? "installed" : "missing"}${summary.ghCli.authenticated ? " (authenticated)" : ""}`,
       `MCP: builtin=${summary.mcpBuiltin.length}, user=${summary.mcpUser.length}`,
     ],