Update for you. I received my timney trigger today. I could not install it on my rifle. long story short, my trigger is out of spec and .005" wider in the rear and doesnt fit in the receiver slot. Timney being the great company they are is taking care of it and getting a replacement out post haste.
However, i did mess with the safety. the factory safety has very little wiggle room in it, but i feel the same play in the seekins one that you do. i’d imagine as you summized earlier, this is due to the design of the berrel and the fact that it is a 60 and 90 degree throw safety where as the factory is not ambi and is a 60 degree only (ICYWY, its the same safety as the SR762). so when i get my replacement trigger, ill install it and let you know what happens. i do notice a lot of wiggle in the timney trigger saefty mechanism. if you wiggle the blade inside the slot where the safety selector fits, there is quite a bit of play, that could contribute to what you are seeing.
Also, FWIW, seekins did tell me that they have never tested their safety with the timney trigger at all so they make no claims to compatibility.