The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $\delta$-sequences, follows the path traced years ago by Stanis?aw ?ojasiewicz in the paper [1], so I describe his approach to the problem below.

?ojasiewicz analyzes the problem for functions of one variable, i.e. $n=1$: by using the definition of *change of variables in a distribution* (see for example [2], §1.9 pp. 21-22) and considering the change of variable $y=x_0+\lambda x$, for $ x,x_0,\lambda \in\Bbb R$, i.e.
$$
\begin{split}
T(x_0+\lambda x)&\triangleq \langle T(x_0+\lambda x),\varphi(x)\rangle\\
&=\left\langle T(y),\frac{\varphi\big(\lambda^{-1} (y-x_0)\big)}{\lambda}\right\rangle
\end{split}
\quad \varphi\in\mathscr{D}(\Bbb R)
$$
he defines the *limit of a distribution at a point $x_o$* as ([1], §1 p. 2-3)
$$
\lim_{x\to x_0} T\triangleq \lim_{\lambda\to 0} T(x_0+\lambda x)
\label{1}\tag{1}
$$
and proves that

- $\lim_{x\to x_0} T=\lim_{x\to x_0^+} T=\lim_{x\to x_0^-} T$
- by using an earlier result of Ziele?ny, if the limit \eqref{1} exists, it is necessarily a constant $C\in \Bbb C$, or more precisely a
*constant distribution*$C$. - a necessary and sufficient condition for the limit \eqref{1} to exist is (see [1], §2, theorem 2.2, pp. 5-7) that $T=f^{(n)}$, where $f\in C^0(\Bbb R)$ and $$ \lim_{x\to x_0}\frac{f(x)}{(x-x_0)^n}=\frac{C}{n!}. $$

Then ** ?ojasiewicz assumes \eqref{1} as the definition of the value of a distribution at a point**: note that this definition does not rely on any particular test function (or sequence of such) $\varphi\in\mathscr{D}(\Bbb R)$, as stated above. Now a few observations:

- ?ojasiewicz ([1], §1 p. 1) states that the case $n>1$ will be analyzed in a subsequent paper which to my knowledge has never been published. However (but this only my opinion), a generalization of \eqref{1} could perhaps be tried by using the Stoltz condition as described, for example, in the textbook of Griffith Bailey Price (1984)
*Multivariable Analysis*, Springer-Verlag. - ?ojasiewicz gives another necessary and sufficient condition for the limit \eqref{1} to exists, in terms of Denjoy differentials ([1], §2, corollary to theorem 2.2, p. 7).
- The term $\lambda^{-1}$, more or less intrinsically used in \eqref{1}, suggests the possible use of the Mellin transform: this suggestion was followed by Bogdan Ziemian in [3], §12 pp. 41-42. He defines a
*(generalized) spectral value of a function/distributions at a point*and proves ([3], §12 p. 43) that it coincides with ?ojasiewicz point value \eqref{1} when this exists (by using the necessary and sufficient condition above): the construction of Ziemaian however does not apply to all distributions.

[1] Stanis?aw ?ojasiewicz (1957-1958), "Sur la valeur et la limite d'une distribution en un point" (French), Studia Mathematica, Vol. 16, Issue 1, pp. 1-36, MR0087905 Zbl 0086.09405.

[2] V. S. Vladimirov (2002), *Methods of the theory of generalized functions*, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029.

[3] Bogdan Ziemian (1988), "Taylor formula for distributions", Rozprawy Matematyczne 264, pp. 56, ISBN 83-01-07898-7, ISSN 0012-3862, MR0931848, Zbl 0685.46025.