I received a similar problem 4 days into buying my new MacBook Pro 16 inch M1 Pro: The MagSafe just keeps on blinking on and off whenever I charge the MacBook when it is completely shutdown. My guess, from what I've read from this thread, is that it is a problem with the Monterey 12.0.01 OS, as I didn't receive the problem when I took the Macbook Pro out of the box. So instead of completely returning it to the store, I decided to be a bit geeky and factory reset the entire device & reinstalled MacOS Monterey in the MacOS Recovery panel (which for M1 macbooks can be activated by pressing and holding the power button when booting up). My Mac reinstalled Monterey 12.0.01, but surprisingly, the MagSafe blinking issue was now gone. I can just plop the MagSafe in, and the thing charges seamlessly.
From this whole experience, I can conclude two things: (1) The MagSafe blinking issue is most likely a software issue, and (2) Monterey 12.0.01 might not be the issue here.
Hence, I would strongly recommend that anyone who experiences this issue either factory reset their device, or, for convenience's sake, just spend some extra money and just buy a USB-C to USB-C cable as a substitute for charging until Apple releases a new OS patch that solves the issue (do recognize that USB-C charging will only accept 94W maximum, which means that it will charge slower than the MagSafe).
Hope this helps.