Comment by userbinator

meaning the modem will run in sleep mode even when you are in airplane mode

AFAIK this is not true at least for the Mediatek 65xx and early 67xx platforms; I've analysed the firmware and hardware on those. They actually power off the modem and rest of the RF system when in airplane mode. The modem only boots up and starts searching for a signal when you take it out of airplane mode, which is why it takes a noticeable time (10-30 seconds, depending on how many bands are enabled) to get a signal. If your phone goes from airplane mode to having a signal and immediately capable of calling, then I suspect it's one where the modem is not truly turned off.

I haven't inspected Broadcom, Qualcomm, or Spreadtrum in any detail to say whether they do things differently.

Are there actually smartphones without an IMEI

Look for a "tablet" or anything else without the word "phone" in it if you just want a touchscreen portable computer. An IMEI is obligatory to connect to cellular networks, in much the same way as a MAC address is to Ethernet and WiFi.