Same core video across TikTok, Reels, and Shorts is usually fine if it's your own content. What hurts more is downloading a watermarked copy and re-uploading it, or using the exact same hook and caption everywhere. I upload the original export, tweak the first couple of seconds and the caption per platform, and pick audio inside each app when I can.
If you're posting at different times, a scheduler helps more than doing it by hand in three apps. I built PostOnce.to to crosspost or schedule your own videos to TikTok, Instagram, YouTube, Facebook, X, LinkedIn, Threads, Pinterest, and Bluesky, including reposting your own TikToks without the TikTok watermark. Setup takes about 10 minutes, it can re-publish older posts, and there's a 7-day free trial.
I'd still check each preview before it goes out, especially the crop and opening line.