There are already a variety of good comments and answers from different points of view; I would like to present another perspective, and provide further details.
does there exist any epsilon delta proof, in which we can only show that the delta only exists (as is the minimum condition in the statement) rather than the explicit relation of it with epsilon?
No, as you guessed; (disregarding foundational issues that are bound to pop up eventually) any $\forall\epsilon,\exists\delta$ statement can be in principle turned into a more explicit relation between $\epsilon$ and $\delta$. Of course, in general how explicit the original setup is bounds from above how explicit this relation could be. Indeed, as discussed at Any function with a modulus of continuity proportional to any preassigned $\epsilon>0$ is Lipschitz-continuous, any $\forall\epsilon,\exists\delta$ is equivalent (disregarding foundations) to some $\exists\delta,\forall\epsilon$, at the cost of a jump in the type of $\delta$; more explicitly
$$[\forall\epsilon\in\mathbb{R}_{>0},\exists\delta\in\mathbb{R}_{>0}: P(\epsilon,\delta)\text{ is true }] \iff [\exists \delta\in F(\mathbb{R}_{>0};\mathbb{R}_{>0}),\forall \epsilon\in\mathbb{R}_{>0}: P(\epsilon,\delta(\epsilon)) \text{ is true}].$$
(In this context such a function $\delta$ is not unique; in contexts in which it is unique, $(\implies)$ with uniqueness is an implicit function theorem.)
More generally (in principle) one can convert (disregarding foundations) any
$$\forall \alpha\in A,\forall x_\alpha\in X_\alpha,\exists y\in Y: P(\{x_\alpha\}_{\alpha\in A},y)\text{ is true }$$
to a
$$\exists y\in F\left(\prod_{\alpha\in A}X_\alpha;Y\right),\forall \alpha\in A,\forall x_\alpha\in X_\alpha: P(\{x_\alpha\}_{\alpha\in A},y(\{x_\alpha\}_{\alpha\in A}))\text{ is true }.$$
why not just keep it in the statement of epsilon delta limit that "$\delta$ can be written as some function of $\epsilon $
, $\delta=f(\epsilon)$ with $f: \mathbb{R^+} \to \mathbb{R^+}$ "
This has to do with the scope and use of the statement to be written, which are ultimately based on personal preferences. For instance in the link above a uniform modulus of continuity for a function $f:X\to Y$ between metric spaces is defined as a function $\mathfrak{m}:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$; however one could just as well define a modulus of continuity as a function $\mathfrak{m}:C^0_u(X;Y)\times \mathbb{R}_{>0}\to \mathbb{R}_{>0}$ such that for any $f\in C^0_u(X;Y)$, for any $\epsilon\in\mathbb{R}_{>0}$, and for any $x_1,x_2\in X$:
$$d_X(x_1,x_2)< \mathfrak{m}(f,\epsilon)\implies d_Y(f(x_1),f(x_2))<\epsilon.$$
(This perspective is useful e.g. when talking about equicontinuity.)
(Here $C^0_u$ is the collection of uniformly continuous functions.)
Another alternative is to consider $X$ and $Y$ as metrizable spaces. Define $\widehat{C^0_u}(X;Y)$ as the set of triples $(d_X,d_Y,f)$ such that $d_X$ is a distance function on $X$ compatible with the topology of $X$, $d_Y$ is a distance function on $Y$ compatible with the topology of $Y$, and $f:X\to Y$ is uniformly continuous w/r/t $(d_X,d_Y)$. Then one can define a modulus of continuity as a function $\mathfrak{m}:\widehat{C^0_u}(X;Y)\times \mathbb{R}_{>0}\to \mathbb{R}_{>0}$ such that for any $(d_X,d_Y,f)\in\widehat{C^0_u}(X;Y)$, for any $\epsilon\in\mathbb{R}_{>0}$, and for any $x_1,x_2\in X$:
$$d_X(x_1,x_2)< \mathfrak{m}(d_X,d_Y,f,\epsilon)\implies d_Y(f(x_1),f(x_2))<\epsilon.$$
(This perspective, although somewhat in jest, is useful when there are different distances involved. For instance if $f:X\to \mathbb{R}$ is $\theta$-Hölder with respect to a distance function $d:X\to X\to\mathbb{R}_{\geq0}$, then it is Lipschitz with respect to the distance function $d^\theta:(x_1,x_2)\mapsto d(x_1,x_2)^\theta$, i.e. $\mathfrak{m}(d,f,\epsilon)=\left(\dfrac{\epsilon}{C}\right)^{1/\theta} \implies \mathfrak{m}\left(C d^\theta,f,\epsilon\right)=\epsilon$; see the discussion at the above link; Semmes calls the operation $d\mapsto d^\theta$ the "snowflake functor"; see the reference at Exponents for Hölder functions on metric spaces . Note that pointwise moduli of continuity can be considered to be better suited for this "fixed topology, different distances" perspective.)
(Yet another alternative is to define a modulus of continuity as a function $\mathfrak{m}: \operatorname{Arr}(\operatorname{Met}_u)\times \mathbb{R}_{>0}\to \mathbb{R}_{>0}$, where $\operatorname{Arr}(\operatorname{Met}_u)$ is the collection (category) of all uniformly continuous functions from a metric space to another one...)
Of course it is rude to deconstruct/universalize/invariantize some concept if it's not absolutely necessary for what is to come. E.g. in hyperbolic dynamics certain objects are automatically continuous, almost never differentiable, but to further the theory one needs something more than continuity (see Why Do We Care About Hölder Continuity?). In general, areas like dynamics where matters like stability, approximation, perturbation and deformation naturally introduce more moving parts.
The discussion at Difference between soft analysis and hard analysis is also somewhat relevant.
Finally, to contextualize some of the comments above, fix a function $f:X\to Y$ between metric spaces, and define its "uniform $\epsilon-\delta$ bundle" as
$$E(f)=\{(\epsilon,\delta)\in\mathbb{R}_{>0}^2\,|\, \forall x_1,x_2\in X: d_X(x_1,x_2)<\delta\implies d_Y(f(x_1),f(x_2))<\epsilon\}$$
with (a priori partially defined; Notation for "function from a subset of $X$ into $Y$"? ) projection $\pi:E(f)\rightsquigarrow \mathbb{R}_{>0}$ onto the first coordinate. Then ($\pi$ is defined everywhere and) there is a global section of $\pi:E(f)\to \mathbb{R}_{>0}$ iff $f$ is uniformly continuous, and existence of such a global section is equivalent to the axiom of choice (see https://ncatlab.org/nlab/show/axiom+of+choice), if each fiber is nonempty. Note that with this framework, if there is a section $\delta: \mathbb{R}_{>0}\to E(f)$, any other function $\delta':\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ whose graph is under the graph of $\delta$ will also be a section. Due to other inequalities needed in a given argument one may have a lower bound $\sigma: \mathbb{R}_{>0}\to\mathbb{R}_{>0}$ also, and it might be important, for what is to come, to establish not only a section $\delta$ that stays above $\sigma$, but also one that is measurable/continuous/differentiable etc.. It's clear that this framework too can be universalized in variety of ways.
(Although, as a disclaimer this framework is somewhat misleading; as one can often focus on countably many values for $\epsilon$ (and $\delta$) (in which case countable choice would be sufficient to produce a section), so that the geometry $\pi:E(f)\to \mathbb{R}_{>0}$ seems to suggest is quite flexible.)
