It's important to bear in mind that the equation you're referring to [the time dilation equation (TDE)] is just one special case of the much more important and more general Lorentz tranformation equations.
The original source of the TDE and the LCE is indeed the Lorentz equations. The Lorentz equations are used to derive the time dilation equation (TDE) and the length contraction equation (LCE). But once you have those, they are the simplest way to answer almost all questions in Special Relativity, and the Lorentz equations just muddy the water.
I said: "It (the TDE) can single-handedly resolve the Twin "Paradox"."
And you said:
No, it can't, because the equation only compares two inertial frames at a time. During the turnaround in the twin paradox, somebody's frame is non-inertial.
OK ... I'll explicitly SHOW how to do it!
Take the case where the traveling twin (he) leaves home, right after his birth, at a speed of o.866 ly/y, and gets to his turnaround point when he is 20. Everyone agrees about that. And everyone agrees that he ages another 20 years on the return trip, so everyone agrees that he is 40 when he returns (because his turnaround is instantaneous and he doesn't age at all during that). By using the TDE, he concludes that she ages 10 years on his outbound leg. And SHE uses the TDE to conclude that when he is 20 at the turnaround, she is 40. They disagree. They are BOTH right. That is just the way Special Relativity IS. Get used to it.
Then, he reverses course, and returns to her at the same speed (0.866 ly/y) as on his outbound leg. He gets 20 years older during his return leg (everyone agrees about that), and he says she ages 10 years on his return leg (she doesn't agree ... she says she gets 40 years older during his return leg). So he EXPECTS to find that she is 20 years old at their reunion. But SHE knows that he will be 40 years old at their reunion (THAT'S part of the scenario description, and she knows (from the time dilation equation) that she will be 80 years old then. They MUST agree about their respective ages at the reunion, because they are colocated and face-to-face there. He is surprised to see that she is 80 years old at the reunion, because the TDE told him that she would age half as fast as he does on each of his inertial legs. Where did he go wrong in his expectations? He THOUGHT she'd be 20 at the reunion, but he found her to be 80. When did that additional 60 years of ageing by her occur during his trip, according to him? It couldn't have occurred during EITHER of the two inertial legs of his trip, because the TDE rules those times with an iron fist. The ONLY place that extra 60 years of ageing by her (according to him) could have occurred is during his instantaneous turnaround ... there is no other possible alternative.
In the simple scenario above, there is only one non-inertial instant during his trip. That is what has allowed us to solve the "paradox" ... there is only ONE instant when she could have undergone that instantaneous ageing (according to him). But in more complicated scenarios, there can be multiple such times, and we need to be able to directly compute all of her instantaneous ageings that he says happen. That can be done using Minkowski diagrams, OR easier by using a very simple equation I derived long ago, called "the delta_CADO" equation. "delta_CADO" is the amount of the instantaneous ageing by her, according to him, when he instantaneously changes his velocity wrt her. If the separation between the two twins at some instant is "L" lightyears, and if the instantaneous change in his velocity at that instant is denoted delta_v, where
delta_v = v_after_velocity_change - v_before_velocity_change,
then
delta_CADO = -L * delta_v .
Velocities are POSITIVE when directed AWAY from her, and NEGATIVE when directed TOWARD her.
Very simple, and it works for arbitrarily many velocity changes by the traveler.