Could an aircraft like a Boeing 777 take off with flaps fully deployed?
Given the right weather, and takeoff weight, yes it could leave the ground. Will it be able to attain the regulatory climb gradient? Probably not. Will it be safe in case an engine fails? Most definitely not.

Landing flaps add lots of drag. That's why they are retracted during an aborted landing for example.

On takeoff, too much flaps means you can stop easier if an engine fails. But it also means you need longer distance to accelerate if the decision is to continue. That means the go-speed will be slower than the stop-speed. See: How is minimum runway length related to V1?

There can't be two decision speeds. Limiting the flight to the slower go-speed will significantly limit how much fuel/payload the plane can carry. And of course if an engine fails during the initial climb, the remaining engine won't be able to keep the plane climbing at a safe rate.

Manufacturers test the different configurations in all conditions, such as weight, runway length, winds, etc. After the data is collected, and with some interpolation, the takeoff and climb performance tables are produced and approved by the regulatory bodies. Any configuration used that is not approved would fail the regulatory and safety requirements, and put everyone in great risk.