Is there an all in one messaging app that puts every chat in a single inbox?

Yes, this category exists and it is usually called a unified inbox or an all in one messenger. The good ones load each service as its own real login and stack them in one window with a single unread list. I use Spanbox for exactly the scatter you are describing (WhatsApp, Telegram, Slack, Gmail, Instagram all in one place). It is local-first, so each account runs on your machine rather than through a server, and it is free. Been the single biggest fix for the saw-it-two-days-late problem you mentioned. Older options in the same space are Rambox and Franz if you want to compare. They work, they just got heavier for me once I had a lot loaded. The key thing to check with any of them is whether the service you care about is supported, because the long tail of niche apps varies.

It exists and it is great for the mainstream stuff. One caveat from experience: anything that relies on an unofficial integration can break when the underlying app changes. The ones that use the official web version of each service (so WhatsApp Web, Telegram Web, and so on) tend to be the most stable, so I would bias toward those.

Genuinely life changing once you set it up. I resisted for ages thinking it was gimmicky. Now everything lands in one place and my brain is quieter.

Email client but for chat is exactly the right way to describe it. That is the whole pitch.