I noticed that all of those relay bypass circuits uses a shottky diode in parallel to the relay coil to tame the kickback voltage.
Somebody further up mentioned that he experienced a lag upon relay turn-off, which could well be attributed to a shottky diode.
I had recently done a circuit with a fairly big contactor, which I had fitted with the usual flyback diode (a 1N4007), and the thing just would not turn off in time (upon hitting the end stop switch). I found out this way, that this kind of protection is hardly ever used in current systems. The reason is that there is (in large contactors) a huge energy charge stored in the activation coil, which has to be destroyed in order to remove the charge. Logically, if the voltage is clamped to 0.7V, the charge can only be removed slowly.
This will apply to a small relay only to a smaller extent, but it does, and we do not know what the OP has used.
On some of my builds, like on a speaker turn-on delay and protection, I want the relay to shut off as quick as possible, so I use a (reverse) diode with a 12V Zener in series. If I monitor the mains failing, such a relay turns off before the mains switch can make a pop in the power amp. This makes a remarkable difference in turn-off speed.
Again, this may be an overkill for such a small relay, but using a shottky diode that clamps at 0.25V is not justified. Nothing is endangered with 0.7V of reverse voltage. Unless there were some other reason I fail to recognize.
BTW for the above contactor system I finally used a 50V MOV, and this shuts down the contactor lightning fast.