Remove references to withSentry function in with-sentry example (#41326)

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## Documentation / Examples

- [ ] Make sure the linting passes by running `pnpm lint`
- [x] The "examples guidelines" are followed from [our contributing
doc](https://github.com/vercel/next.js/blob/canary/contributing/examples/adding-examples.md)

## Content

In version
[`7.15.0`](https://github.com/getsentry/sentry-javascript/releases/tag/7.15.0)
of the `@sentry/nextjs` SDK we introduced a new build-time feature that
removes the need for manually wrapping API routes.

This PR removes all manual `withSentry` calls from the `with-sentry`
example.

Co-authored-by: Balázs Orbán <info@balazsorban.com>

examples/with-sentry/README.md

@@ -7,7 +7,6 @@ This is an example showing how to use [Sentry](https://sentry.io) to catch and r
 - `sentry.server.config.js` and `sentry.client.config.js` are used to configure and initialize Sentry
 - `next.config.js` automatically injects Sentry into your app using `withSentryConfig`
 - `_error.js` (which is rendered by Next.js when handling certain types of exceptions) is overridden so those exceptions can be passed along to Sentry
-- Each API route is handled with `withSentry`
 
 ## Preview
 

examples/with-sentry/pages/api/test1.js

@@ -1,10 +1,6 @@
-import { withSentry } from '@sentry/nextjs'
-
 const doAsyncWork = () => Promise.reject(new Error('API Test 1'))
 doAsyncWork()
 
-async function handler(req, res) {
+export default function handler(req, res) {
   res.status(200).json({ name: 'John Doe' })
 }
-
-export default withSentry(handler)

examples/with-sentry/pages/api/test2.js

@@ -1,13 +1,9 @@
-import { withSentry } from '@sentry/nextjs'
-
 function work() {
   throw new Error('API Test 2')
 }
 
 work()
 
-async function handler(req, res) {
+export default function handler(req, res) {
   res.status(200).json({ name: 'John Doe' })
 }
-
-export default withSentry(handler)

examples/with-sentry/pages/api/test3.js

@@ -1,13 +1,9 @@
-import { withSentry } from '@sentry/nextjs'
-
 function work() {
   throw new Error('API Test 3')
 }
 
-async function handler(req, res) {
+export default function handler(req, res) {
   work()
 
   res.status(200).json({ name: 'John Doe' })
 }
-
-export default withSentry(handler)

examples/with-sentry/pages/api/test4.js

@@ -1,16 +1,11 @@
 import * as Sentry from '@sentry/nextjs'
 
-async function handler(req, res) {
+export default function handler(req, res) {
   try {
     throw new Error('API Test 4')
   } catch (error) {
     Sentry.captureException(error)
   }
 
-  // Flushing before returning is necessary if deploying to Vercel, see
-  // https://vercel.com/docs/platform/limits#streaming-responses
-  await Sentry.flush(2000)
   res.status(200).json({ name: 'John Doe' })
 }
-
-export default Sentry.withSentry(handler)