Should I stop the FTMO EA during news events, or just let it keep running?

That is a personal choice. We would stop them only when there are Rate decisions, FOMC, or the essential news.

However, there is a spread protection so if there is news and the spread increases, the EA will not open trades.

Sometimes the volatility is before or after the news, so the most realistic filter is the spread, not exactly to stop trading at the news.