ESP-01 wakes up from deep sleep and then hangs
Not necessarily an answer, but something to look at: check the Manufacturer ID of the Flash on your board.  My suspicion is that when some Flash chips come out of suspend, they're in lower drive mode than on initial power-up.  At least one person fixed his 'zombie mode' by replacing the Flash with a different vendor.  Nobody else has attempted it yet.
I have an Inland ESP-01S that's reporting MID = 0x5E (supposedly Zbit) and it comes out of Deep Sleep repeatedly.  Here's all of the Manufacturer IDs I know of or have seen:
Winbond EF
GigaDevice C8
Puya 85
Bright Moon 53
BergMicro E0
XMC 20
XTX 0B
Eon 1C
Shanghai Fundan A1
Macronix C2
Hua Hong Zhixin 68
ISSI 9D
Paragon E0 (same as Berg - oopsie or clone?)
Zbit 5E
Zetta BA
esptool.py --port /dev/ttyUSB0 flash_id   (linux)
esptool.py -p COMx flash_id    (Windoze)
Note: the cheap D1 Mini boards with no-name ESP-12F modules flooding eBay, Aliexpress and other places report XTX Flash, and they have a 5.6K pull-up on EXT_RSTB which will shorten the RST pulse from GPIO16, as well as making it hard to hit the level to be detected as a LOW due to the series 470 ohm resistor.  We're unsure whether it's the Flash or the low value pull-up (typically 10 or 12K on other boards/modules) that's causing issues with those boards.  Normally if RST isn't seen as LOW after Deep Sleep then you'll get an &quot;ets_main.c&quot; message at boot before it hangs.
I have a clone D1 Mini Pro that has a Flash that reports MID = EF (Winbond), although the marking on the Flash is nothing I can find on Winbond's site, only on places selling cheap Flash.  Luckily the D1 Mini Pro also exits Deep Sleep successfully.