Why didn’t the Concorde have flap-canards like the Tu-144?
It did not need them. Concorde could operate from most commercial airports and did not need especially long runways. Retractable canards add drag and the mechanism is a point of failure.

I fail to see how Tu-144 canards is a “superior” feature as they were added to support a less effective wing.

Very much related: Why is the Tu-144 the only commercial airplane with canard configuration?