I may allow degrees of success in the 5e game, similar to what you are saying in the blades game. In the example, I may set the bribe DC at 15 and if the PC rolls the 15 then all is good. If the PC rolls 20 then the guard may not take a bribe and simply let the PC pass while if the roll is 12, the guard may ask for a higher bribe. Fail by 5 or more you may get attacked or have other guards called.Let me compare and contrast how a bribery attempt works in 5e versus Blades. Hopefully this will showcase curation. First, let's say both scenarios show up unexpectedly, and the GM is doing some quick ad-lib. A roguish sort of PC has been caught up in a compromising position by a guard, and is offering a sizable bribe to escape custody.
In 5e, the DM will consider how the guards in this area fit into his campaign -- is it largely law and order, or is it more corrupt. Then, the GM will consider how this guard thinks, and compare to the amount offered This will set the DC for the attempt (or if the attempt is even allowed). A roll is called for, and if the PC succeeds, the guard accepts the bribe, if the PC fails, the guard refuses the bribe.
In Blades, the GM considers the situation - is the PC acting in a controlled situation, a normal one, or a is the PC desperate. Given that the PC just got caught, the GM will likely say this is a desperate situation. Next, the GM looks at the guard's tier (set by setting, but the Crew knows this because they got to pick the target). This compared to the PC's tier will set the effect level -- let's assume normal (even). Then the PC says what attribute their rolling (they get to pick) and a roll is made. On a success, this guard is bribeable, and the transaction occurs. On a partial, the situation is good and bad -- the guard may be bribeable but haggle for a higher bribe. On a failure, the guard isn't bribeable, or another guard shows up, or the PC accidentally insults the guard's mom -- something bad happens