Yhyjyk

Does there exist a countable set of connected proper smooth $mathbbC$-schemes such that any connected proper smooth $mathbbC$-scheme admits a $mathbbC$-immersion into one of them?

I am just posting my comment as one answer. So long as you are only asking about schemes (rather than complex analytic spaces), you can avoid the hard analysis from the previous MathOverflow answer. The argument below sketches this. The main additional detail beyond Hilbert scheme techniques is a strong variant of Chow's Lemma.

Chow's Lemma. Let $X$ be a separated, finitely presented scheme over a field $k$, and let $Usubset X$ be a dense open subscheme that is a quasi-projective $k$-scheme. There exists a strongly projective morphism $nu:widetildeXto X$ such that $widetildeX$ is a quasi-projective $k$-scheme and such that the restriction of $nu$ over $U$ is an isomorphism.

There may be an earlier source, but the source that I know is the following article.

MR0308104 (46 #7219)

Raynaud, Michel; Gruson, Laurent

Critères de platitude et de projectivité. Techniques de "platification'' d'un module.

Invent. Math. 13 (1971), 1–89.

Finally, the very last step of the argument requires Nagata compactification.

Nagata compactification Every separated, finite type $k$-scheme is isomorphic to a dense open subscheme of a proper $k$-scheme.

For every separated morphism that restricts as an isomorphism over a dense open in the target, the isomorphism locus is the maximal open subscheme of the target over which the morphism is an isomorphism.

Corollary. For every proper $k$-scheme $X$, there exist integers $n,mgeq 1$ and there exists an $n$-tuple of pairs of $k$-morphisms, $$(nu_ell:widetildeX_ellto X,e_ell:widetildeXhookrightarrow mathbbP^m_k)_ell=1,dots,n,$$ such that the $nu_ell$ are strongly projective $k$-morphisms whose isomorphism loci cover $X$ and such that the morphisms $e_ell$ are closed immersions of $k$-schemes.

Proof. Since $X$ is quasi-compact, there exists a finite covering of $X$ by open affine subschemes. By Chow's Lemma, for each open affine, there exists a strongly projective $k$-morphism from a projective $k$-scheme to $X$ whose isomorphism locus contains that open affine. QED

From now on assume that $k=mathbbC$. Using Hironaka's Theorem, there exists such an ordered $n$-tuple with every $widetildeX_ell$ a smooth, projective $mathbbC$-scheme. Call such an ordered $n$-tuple a smooth Chow covering.
Smoothness is convenient, because we can use Zariski's Main Theorem: the closed complement in $widetildeX_ell$ of the isomorphism locus is the union of all positive-dimensional components of fibers of $nu_ell$. Stated differently, for the associated closed subscheme $Y_ell,ell:=widetildeX_ell times_X widetildeX_ell$ with its two projections to $widetildeX_ell$, the isomorphism locus of $nu_ell$ equals the isomorphism locus of these two projections to $widetildeX_ell$.

For every smooth Chow covering, for every $1leq j,ellleq n$, denote by $Y_j,ell$ the closed subscheme $widetildeX_jtimes_X widetildeX_ell$ of the fiber product $widetildeX_j times_textSpec mathbbC widetildeX_ell$. For every $ell$, denote the diagonal morphism by $$delta_ell:widetildeX_ell to Y_ell,ell.$$ For every $(j,ell)$, denote by $$sigma_j,ell:Y_j,ell to Y_ell,j$$ the isomorphism that transposes factors. For every ordered triple $(j,ell,r)$, denote by $$c_j,ell,r:Y_j,elltimes_widetildeX_ell Y_ell,r to Y_j,r,$$ the morphism induced by the first and final projections. Altogether, the Chow datum of the smooth Chow covering is the collection, $$(([widetildeX_ell])_ell,([Y_j,ell])_j,ell,([delta_ell])_ell,([sigma_j,ell])_j,ell, ([c_j,ell,r])_j,ell,r).$$ The first two parts of the datum are points in appropriate Hilbert schemes. The last three parts are points in appropriate Hom schemes.

There are a number of compatibilities satisfied by the part of a Chow datum of a smooth Chow covering. Each of these compatibilities defines a locally closed subscheme of the Hilbert / Hom scheme parameterizing the Chow datum. The main compatibility is that for every $ell$, the images in $widetildeX_ell$ of the inverse images in $Y_j,ell$ of the isomorphism locus $U_j$ in $widetildeX_j$ (defined with respect to $Y_j,jto widetildeX_j$) give a covering of $widetildeX_ell$. This is what we need to glue together the open subschemes $U_ell$ to form a smooth scheme $X$ together with morphisms $nu_ell:widetildeX_ell to X$ whose fiber products equal the datum $([Y_j,ell])_j,ell$.

Altogether, there are countably many ordered pairs $(n,m)in mathbbZ_geq 0times mathbbZ_geq 0$. For each such pair, there are countably many ordered $n$-tuples of Hilbert polynomials for the closed subschemes $widetildeX_ell$ of $mathbbP^m_k$. For each such ordered $n$-tuple of Hilbert polynomials, for the associated $n$-fold product of Hilbert schemes parameterizing $(widetildeX_ell)_ell=1,dots,n$, there are countably many $n^2$-tuples of Hilbert polynomials for the closed subschemes $Y_j,ell$. For each such tuple, there is a further relative Hilbert scheme parameterizing closed subschemes $Y_j,ell$ in $widetildeX_jtimes_textSpec mathbbCwidetildeX_ell$. Then there are countably many components of each Hom scheme for the morphisms $delta_ell$, $sigma_j,ell$ and $c_j,ell,r$. For each of these countably many quasi-projective $k$-scheme parameterizing a Chow datum of a smooth Chow covering of a smooth $k$-scheme, we glue together $X$ and the morphisms $nu_ell$ from the isomorphism loci.

In the end, for each of the countably many data, $iin I$, of $(m,n)$ and all of the relevant Hilbert polynomials, there is a quasi-projective $mathbbC$-scheme (with its induced reduced scheme structure), $B_i$, and a datum, $$(pi_i:mathcalX_i to B_i, (widetildemathcalX_i,ell subset mathcalX_itimes_textSpec mathbbC mathbbP^m_mathbbC)_ell)$$ of a smooth, proper morphism $pi_i$ and a collection of closed subschemes $widetildemathcalX_i,ell$ such that each projection, $$widetildemathcalX_i,ell to B_itimes_textSpec mathbbCmathbbP^m_mathbbC,$$ is a closed immersion of a smooth $B_i$-scheme into projective space over $B_i$, and such that the projections, $$(widetildemathcalX_i,ellto mathcalX_i)_ell=1,dots,n$$ define a smooth Chow covering. Every smooth, proper $mathbbC$-scheme occurs as a fiber of some $pi_i$.

Finally, if you insist that each $B_i$ be smooth, you can always arrange that by applying Hironaka's Theorem to each $B_i$. The effect is that every $mathcalX_i$ is itself a separated, quasi-compact, smooth $mathbbC$-scheme. It certainly is not proper. However, we can apply Nagata compactification and Hironaka's Theorem (for a third time) to realize $mathcalX_i$ as a dense open subscheme of a proper, smooth $mathbbC$-scheme.