This sounds very similar to a bug with older BIOS on this exact board model (11th gen i5/i7). It would happen when the battery was empty, which is consistent with a delay between cracking your screen and having it replaced before trying to boot again.
If you’re using a BIOS with version before 3.07, this may be the case.
Unfortunately, you can’t upgrade the BIOS if the system won’t turn on… you will need to fully reset the mainboard state.
Also check the Community BIOS guide, there is a section for the 11th gen FW 13.
Edit: as an additional detail, older HDMI and other modules will drain the battery faster that would be usually expected, leading to frequent complete draining of the battery. If you manage to reset the mainboard, consider upgrading some of your modules depending on what you have since initially getting the laptop, and upgrading the BIOS to the latest. Carefully review the changelog too.