Providing voltage to USB port and battery terminals from same voltage regulator. Safe to do?
First of all, please, limit your battery voltage to 4.2. Batteries operate at 3V-4.2V (lower boundary can vary, 2.8-3.3), so anything from 3.5 to 4.2 will be 100% safe. I fix the phones from time to time, so I connect the power supply directly to the battery pins all the time :)
Excessive battery voltage, even if not much, can be harmful in the long run. One day some 3 months away from now it won't boot. Or maybe everything will be OK. In any case, it's an unnecessary risk you can easily remove.
Second of all, in order for the phone to boot, it only needs to &quot;feel&quot; the usb voltage just for a moment, and the internal electronics will be like HEY BRO TIME TO WAKE UP. Assuming you have already changed your voltage to 4V (which I strongly recommend), try to apply 4V to the charging port for a second and see if it starts to boot. If so, you can do the same trick as well - you can implement some short 4V pulse on the usb power line.
But KEEPING usb voltage at 4V is a BAD idea. The circuit is intended to step 5V input down to charging voltage (which varies within battery voltages, up to 4.2V). Too small difference between voltage input and output can cause all sorts of trouble. You only care about HEY WAKE UP signal.
All of this is based on assumption that 4V on USB power line will be enough for the phone to start booting. In any case I strongly advise AGAINST charging and powering battery terminals from the power supplies: battery terminal not only gives energy to the phone, but also received it when charging, and constant presence of USB will simply try to &quot;charge&quot; your battery terminal power supply, which is very likely not designed to sink current. CORRECTION: it won't sink current because usb and &quot;battery&quot; voltage is the same, but it's still not designed to have battery and charger voltage on the same level
So you need to create a pulse on the USB line. either 5v or if you're lucky 4V will be enough. Now it depends on how much into electronics you are. Making some small circuit with RC timer and a comparator can make it easy to make 1-2s pulse on USB and then go low, for instance. Mechanical switch is always an option. A microcontroller can open and close mosfet on USB power line too. Handful of options.
By the way, since you're inside the phone, you can actually wire yourself up to the power button. All you need is to pull the power button dataline down for 3 seconds and your phone will start to boot. You can easily start your phone with tweezers if you know what and where exactly to short to the ground for a moment. Nothing stops you from using it too, in fact, it would be a pretty elegant solution in my opinion, and it doesn't care about voltages too!