If you want to check on the compatibility of Shimano components then you can do that at this link...

2023-2024 SHIMANO Compatibility

It might be a little confusing at first, but you'll figure it out. This is the current compatibility list and shows the components currently in production and perhaps a previous version or two in some cases. So you can figure out what Shimano components will work with other series Shimano components. At page index C-499 you can find the ST-4720 is compatible with the BR-4770. As to whether there is any difference that you need to consider for diameter of your current rotors or the mounting of the calipers, you'll have to dig more into that. I'd think mounting is the same, but rotor size that's recommended might be different.

The line up chart will also help you see what components are suggested to go with a particular series or group set.

2023-2024 SHIMANO Line-Up chart

There is a link at the top of the page for both charts to get into the archives if you ever want to look at such for older components.

And this link will take you to the main page of the Shimano Tech doc site where you can find everything they make publicly available. The DM's are the most useful for component installation. But they didn't start making DM's until later in this Century.

Manuals | Manuals & Technical Documents (shimano.com)

I'd still make certain that you have your current brakes working properly. I'm not sure I quite agree that hydraulic is significantly better at stopping you or will in all cases be one-finger use. Most all brakes I've used will, when adjusted well and in peak condition, lock up the wheel. Albeit some can be difficult to achieve this. But locking up the wheel and skidding isn't good for fast stopping. But at least shows that they are able to give the maximum braking that the tires on the wheels can give you if modulated correctly.

If it's worth it to you for whatever reason, then that's fine. If nothing else, you'll get what you want and some experience DIY'ing.