The first thought isn't about repair. It's about the photos. Summer, the kids, the parents, the cat that fell asleep on the keyboard in such a funny way. All of it is sitting in the phone's memory, and right now it feels like those pictures are out of reach.
In reality, things with the data aren't as bad as they look from the outside. A broken screen is damage to the display, not necessarily to the memory. The photos themselves are stored on the drive, and in most cases after a screen replacement the phone simply turns on and shows everything that was there. Sometimes it takes a bit more effort: for example, if the phone doesn't respond to touch, the technician connects it to equipment and pulls the data directly, without touching the broken digitizer.
Where do they start in cases like this? With diagnostics. The technician checks whether the phone itself is alive under its broken exterior: whether the motherboard responds, whether the device charges, whether the tooling recognises it. Until there's a clear picture of what's going on inside, no one is going to randomly swap parts on a working device that holds important data.
The main rule that people forget: don't try to "check" the phone after a drop by taking it apart with whatever's at hand. Any unnecessary intervention at this stage — a hairdryer, a screwdriver from an eyeglass kit, an attempt to bend the frame back — can turn a simple recovery into a long and sad process. It's like opening a safe with a crowbar: whatever's inside is more likely to be damaged by your hands than by the original accident.
So if the screen is broken and a whole photo chronicle lives inside, leave the phone alone and let it get to someone who knows how to work with stories like this. Your job is simply not to hit "erase all" in a panic and not to plug it into the first laptop you come across in the hope that it'll "sort itself out". The data will survive both the fall and the cracks — as long as you don't get in the way of the people who'll be getting it out.