Magsafe 3 blinking orange slowly and Mac not turning on

The original post you tagged your question 14" 2021 MacBook Pro was a person dealing with likely a failed battery which is why we were disconnecting the battery to be sure. The older 13" 2020 MacBook Pro post you linked to wanted to run their system without a battery!

Two very different directions! Disconnecting the battery to validate a bad battery is not the same as running a system as a usable system without a battery!

Think it this way... your cars brakes are soft and you don't know if the power brake pump is working or the brakes them selves are worn. While a real mechanic would put your car on a lift pulling the wheels off to inspect the brakes, you don't have a way to do that so you reach under the hood checking the brake fluid and maybe check the pressure with a gage. But of course you wouldn't drive with the brake line not connected right?

So... why is the MagSafe blinking Orange?

Whenever I try to charge my system the Magsafe 3 LED blinks slowly Orange and even if I leave it for 30+ min it doesn't turn on. I have tried with USB-C charging as well and it doesn't turn on.

If it's doing the same, then your Battery has likely failed! It's time for a new battery! MacBook Pro 16" 2021 Battery Replacement

If your system is still running see if you can grab a screenshot of this Apps main window, it's by far better informationally than what Apple offers in their macOS CoconutBattery post it here so we can see it.

And if your system is not turning on you should be able to disconnect the batteries power connection and still using the MagSafe port connection the system should respond and as I stated I haven't serviced this 2021 series batteries yet. And yes, it's the start of them failing as it's not just cycles it's also age.

But once again, this is not a safe setup to run your system! And lastly, neither the MagSafe, or using the USB-C connection is not foolproof! Trying to run stuff on your system as a workable system without the Battery Is playing Russian Roulette! With a sizable risk of file storage corruption.