ccl/sqlproxyccl: fix TestConnectionMigration test flake

Fixes #83096.

It appears that database/sql does not provide any thread-safe guarantees for
methods on the Conn type except the Close method. This commit fixes a test-only
issue that causes a panic when there's a race between internal Conn methods
by ensuring that Conn methods are used in a single-threaded way.

Release note: None

Release justification: sqlproxy only test change.

pkg/ccl/sqlproxyccl/proxy_handler_test.go

@@ -1230,10 +1230,8 @@ func TestConnectionMigration(t *testing.T) {
 		conn, err := db.Conn(tCtx)
 		require.NoError(t, err)
 
-		// Spin up a goroutine to trigger the initial connection.
-		go func() {
-			_ = conn.PingContext(tCtx)
-		}()
+		// Trigger the initial connection.
+		require.NoError(t, conn.PingContext(tCtx))
 
 		var f *forwarder
 		require.Eventually(t, func() bool {
@@ -1283,16 +1281,10 @@ func TestConnectionMigration(t *testing.T) {
 		// one test.
 		<-goCh
 		time.Sleep(2 * time.Second)
-		// This should be an error because the transfer timed out.
+		// This should be an error because the transfer timed out. Connection
+		// should automatically be closed.
 		require.Error(t, f.TransferConnection())
 
-		// Connection should be closed because this is a non-recoverable error,
-		// i.e. timeout after sending the request, but before fully receiving
-		// its response.
-		err = conn.PingContext(tCtx)
-		require.Error(t, err)
-		require.Regexp(t, "(closed|bad connection)", err.Error())
-
 		select {
 		case <-time.After(10 * time.Second):
 			t.Fatalf("require that pg_sleep query terminates")