Why did the on-screen home button vanish when upgrading to 14.5.1 on iPhone SE 1st generation?

I think you mean the assistive touch. Her is how you can enable it: Use AssistiveTouch on your iPhone, iPad, or iPod touch

1. Go to Settings > Accessibility > Touch, then select AssistiveTouch to turn it on.
2. Use "Hey Siri" to say, “Turn on AssistiveTouch.”
3. Go to Settings > Accessibility > Accessibility Shortcut and turn on AssistiveTouch.

I'm not sure what you mean by "on screen home button" on a SE 1st gen. The SE has never had an on screen home button, it's built into the phone case below the screen. The only app that gets an on screen button is the camera, but that's the shutter button, not a "home" button.

I'm using SE1 and 14.5.1 too

You can customize it.

Changing A Button

You don’t necessarily have to remove a button to replace it with a new one. You can edit a button so that it is replaced with a different one. To do so, tap the button you want to replace on the customization screen. A list of all supported functions will appear. Select which function you want to swap the old one out for and tap ‘Done’.

A Little Tip

As we mentioned earlier, most users become familiar with AssistiveTouch when their Home button stops working. They don’t really need any of the other buttons that appear when you tap the AssistiveTouch button. If you want the AssistiveTouch button to only execute the Home button’s function when you tap it, remove all other buttons from the menu as described above. Keep the Home button and when you tap the AssisitveTouch button, it will execute the Home button’s function without showing you a menu of other buttons.