Your terminology isn't quite standard, and might lead to confusion in more complicated scenarios. For example, Winer's "Statistical Principles in Experimental Design" (2nd edition, McGraw-Hill, 1971), defines "main effects" for each level of a categorical variable, expressed as the associated difference from the grand mean. What you describe are called "differential main effects" in that text, the difference between two individual "main effects" within a categorical variable (p. 316).
With only 2 levels of a categorical predictor A, a test of the "differential main effect" between its two levels is equivalent to what you seek as a test of the overall "main effect" of A. With more levels, however, you need to combine all the "main effects" or "differential main effects" to evaluate the overall significance of A.
I would get lost if I tried to present the interaction terms in the way that you show. Similar to the handling of "main effects," the classic treatment in the Winer text defines interaction effects for each combination of levels of the interacting predictors, and then goes on to define "differential interaction effects" and "simple interaction effects," which I don't find very simple.
There are several ways to code categorical predictors for a model. Several are explained on this page. The comparisons you display are close to what you might get with deviation coding. Also explained on that page, all codings lead to the same model. The model-coefficient values will differ depending on coding, but all models will return the same predictions and the same inference about the predictors.
I find it simplest to think about interactions when the categorical predictors are dummy coded (also called "treatment coded"), the default in R. One of its $k$ levels is chosen as the reference, and the categorical predictor is then described as a set of ($k-1$) values of "0" or "1," with a "1" representing which (if any) non-reference level is in place for an observation. Then your 3-way ANOVA (with outcome y and dummy-coded binary predictors A, B and C) can be represented as the following linear model with 8 regression coefficients ($\beta$s):
y ~ $\beta_0$ + $\beta_a$ A + $\beta_b$ B + $\beta_c$ C +
$\beta_{ab}$ AB + $\beta_{ac}$ AC + $\beta_{bc}$ BC +
$\beta_{abc}$ ABC
Let's say that A1, B1, and C1 are the reference levels. Then each of A, B, and C in the above formula has a value of 1 only when the corresponding categorical predictor is not at its reference level. In an interaction (product) term, if any predictor in the term is at its reference, that whole term is thus 0.
In that coding, each coefficient has a simple interpretation: a difference from what would be predicted from lower-level coefficients. In your terminology with a completely balanced design:
$\beta_0$, the intercept, is $\bar{X}_{T_1}$: all predictors at their reference levels.
$\beta_a$ is $\bar{X}_{T_5}-\bar{X}_{T_1}$, the difference from the intercept associated with non-reference A (dummy-coded value of 1), with the other predictors maintained at reference (dummy-coded values of 0). Similarly for the other single-predictor coefficients: $\beta_b$ is $\bar{X}_{T_3}-\bar{X}_{T_1}$ (A and C at reference) and $\beta_c$ is $\bar{X}_{T_2}-\bar{X}_{T_1}$ (A and B at reference).
$\beta_{ab}$ is the extra difference from what would be predicted based solely on those individual coefficients, with C maintained at reference: $\bar{X}_{T_7}-\beta_0-\beta_a-\beta_b$ Substituting the above gives $$\beta_{ab}=\bar{X}_{T_7} -(\bar{X}_{T_1})-(\bar{X}_{T_5}-\bar{X}_{T_1})-(\bar{X}_{T_3}-\bar{X}_{T_1}) \\
= \bar{X}_{T_7} - \bar{X}_{T_5}-\bar{X}_{T_3}+\bar{X}_{T_1}.$$
Similarly, $\beta_{ac}$ is $\bar{X}_{T_6} - \bar{X}_{T_5}-\bar{X}_{T_2}+\bar{X}_{T_1} $. $\beta_{bc}$ is $\bar{X}_{T_4} - \bar{X}_{T_3}-\bar{X}_{T_2}+\bar{X}_{T_1}$.
Finally, the 3-way interaction coefficient is the difference between $\bar{X}_{T_8}$ (mean outcome with all predictors at non-reference) and what you would have predicted based on all the lower-level coefficients:
$$\beta_{abc} = \bar{X}_{T_8} - \beta_0 - \beta_a -\beta_b - \beta_{ab} - \beta_{ac}-\beta_{bc}$$
Substituting gives:
$$\beta_{abc} = \bar{X}_{T_8} - \bar{X}_{T_7} + \bar{X}_{T_5} - \bar{X}_{T_6} + \bar{X}_{T_2} - \bar{X}_{T_4} + \bar{X}_{T_3} .$$
Those coefficients based on dummy coding are much easier to associate with cell means than what we might have gotten if I tried to carry through your formulation. More generally, in situations with unbalanced designs or generalized linear models, the cell means used above would instead be the modeled estimates for the indicated (combinations of) predictors.
The downside of dummy coding is that there is then a temptation to consider, for example, $\beta_a$ as the "main effect" coefficient for A. That's incorrect: it only represents the difference from the intercept associated with the shift from A1 -> A2 when B and C are at reference levels (B1,C1). Even the 2-way interaction coefficients are only for the situation when the predictor omitted from the interaction is at its reference.
Tools for post-modeling calculations allow evaluation of overall associations of predictors with outcome. For example, with a balanced design you can perform a linear regression model with dummy-coded predictors and use the basic R anova() function on the model to get an analysis of variance table.
Final note:
I'm pretty sure that if I had managed to work through the 3-way interaction coefficient in the way you requested, I would have come up with the same final result. An interaction in a factorial design is the extent to which the result for a treatment combination cannot be predicted from the corresponding lower-level "main" and "interaction" effects. Although I worked this through with dummy coding, the outcome with all 3 predictors at non-reference levels is still $\bar{X}_{T_8}$ and the remaining terms in the formula for $\beta_{abc}$ come from what would be predicted based on the lower-level "main" and "interaction"effects. The net combination should add up the same however the predictors were coded.
