(The strikethrough'd text (and other mentions of "backlight" below) is irrelevant because I forgot to consider this was the generation of iPhone that switched to OLED. See the explanation from Jerry Wheeler below. I am leaving it here for posterity but do not rely on this advice unless your screen was replaced with a cheaper aftermarket LCD screen.)
Are you completely sure that the screen is BLACK and not that the backlight is off but the screen is displaying an image? To test this go into a dark room and shine a bright flashlight (not a cell phone ideally, if you have it a flashlight is better) at the screen from different angles. Can you see the image kind of "ghost reflected" in the light of the flashlight?
If this is the case, it is a backlight not turning on. If the entire screen has no image, it likely means the chip that processes images on the logic board is fried and needs replacing. (See last paragraph.)
Since the backlight works (i.e. you stated the apple logo will be lit up), if it is simply not turning on to help you see the image then it is a different problem. It could be a ambient light sensor issue, or it could be a failure of a chip on the motherboard, or it could be yet another failed and faulty install of iOS 16. Alternatively, there is a small (but not zero) chance the backlight is failing and only has enough "oomph" to turn on for a second or two during boot up before it overheats a component and just can't do it anymore.
Depending on where your situation is with data and backups and such, it may be worth it to try either hardware repairs (simply disconnecting the earpiece speaker can help diagnose the first option), or deleting data and resetting the device to get a freshly reinstalled version of your current iOS (since Apple unsigns all old iOS once you've upgraded, you cannot ever go back to an older stable build without violating ToS), and that could possibly fix the issue if it is a bad install.
Finally, if you have access to a new screen it wouldn't hurt to try replacing the screen just in case it is that unlikely strange backlilght failure.
If all else fails, that iOS update probably fried a chip on your logic board that controls the backlight so it never gets the proper signal to turn on and control the backlight part of the screen. A phone repair is possible at a dedicated board repair specialist, but it is definitely the kind of thing that most users here (myself included) cannot do. You can find a dedicated board repair specialist in your area, though be advised that it often costs more than the phone is worth to get it repaired. MDRepairs on YouTube has many short-form videos, and Louis Rossman has long-form videos with amazing explanation and questions answered, of these types of repairs, so you can get an idea of the work involved and why it costs so much.
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