I thought it was handled extremely well actually. Leonard Snart has the secret yes, but given he knows The Flash would never allow himself to be blackmailed or coerced the dillema is what he can do with the knowledge, he could threaten Barry's frinds and family when needed but this would just force Barry's hand and he either calls Snarts bluff on the matter or gets beaten to pulp and 'dissapears' courtesy of a vengeful Flash. Really it's his secret, and yet his options for what to do with it are near non-existent realistically. So he keeps it his secret, his alone. Until the time he needs something from Barry/The Flash....
Exactly. Snart seems convinced that Flash will kill him and his crew if he leaks Barry's ID.
To be honest I would not call Barry's bluff since given how fast Barry can move he could easily beat Snart to a pulp before Snart could blink his eyes.
Few people are more dangerous then when they are defending their loved ones and I think Snart understands that about Barry.