It is easier to think about the flatness problem in the other direction: from the past to the present. First, we need to introduce some quantities: $$E \equiv \frac{H}{H_0},\qquad \bar{\Omega} \equiv \Omega E^2, \qquad \bar{\Omega}_k \equiv \Omega_k E^2, \qquad \Omega_k \equiv -\frac{k}{a^2H^2},$$ where $H_0$ refers to the the <PERSON> constant today, i.e., $H_0 \equiv H(a_0)$ and $a_0$ representing the scale factor today. In these variables $\bar\Omega + \bar\Omega_k = E^2$, $\bar\Omega(a_0) = \Omega(a_0)$ and $\bar\Omega_k(a_0) = \Omega_k(a_0)$.
It is easy to see that $\Omega \propto \rho$, and therefore, for a fluid with arbitrary constant equation of state $w$, $$\bar\Omega_w = \Omega_{w0}\left(\frac{a_0}{a}\right)^{3(1+w)},\qquad \Omega_{w0} \equiv \Omega_w(a_0).$$ If the only component of the universe is the $w$ fluid, then, $\Omega = \Omega_w$. To simplify it further, lets choose a dust like fluid $\Omega_m$ with $w = 0$ (however, the argument bellow is valid for any combination of components with equations of state larger than $-1/3$).
Now, lets say that initially $\vert\Omega-1\vert$ is very close to one, that is, for an initial value of the scale factor $a_i$, we have $$\Omega_{mi} \equiv \Omega_m(a_i) = 1-10^{-5}, \qquad \Omega_{ki} \equiv \Omega_k(a_i) = 10^{-5}.$$ In terms of these initial densities we have, $$\Omega_m = \frac{\Omega_{mi}(a_i/a)^{3}}{\Omega_{mi}(a_i/a)^{3} + \Omega_{ki}(a_i/a)^{2}} = \frac{1}{1+\alpha a / a_i}, \quad \Omega_k = \frac{\Omega_{ki}(a_i/a)^{2}}{\Omega_{mi}(a_i/a)^{3} + \Omega_{ki}(a_i/a)^{2}} = \frac{\alpha a / a_i}{1+\alpha a / a_i},$$ where we defined $\alpha = \Omega_{ki} / \Omega_{mi}$. 418
Putting these initial conditions around the nucleosynthesis epoch, we have $a_i \approx 10^{-10}a_0$ and consequently $$\Omega_{m0} \approx \frac{1}{1+\alpha 10^{10}} \approx 10^{-5}, \qquad \Omega_{k0} \approx \frac{\alpha 10^{10}}{1+\alpha 10^{10}} \approx 1.$$ The situation is completely reversed, in this case today we would have an universe completely dominated by curvature. To avoid this, we would need $\alpha \ll 10^{-10}$, i.e. $\Omega_{ki} \ll 10^{-10}$.
Now, we can answer your questions:
1. No matter which value Ω(t0) we would measure, it would be arbitrary close to 1 for small enough t
That's true as long as the matter content of the universe has an equation of state larger than $-1/3$. However, that's exactly the problem, since the matter content tends to dominate in the past, the ratio $\Omega_k/\Omega$ is led dynamically to zero, even if $\vert\Omega_{k0}\vert > \vert\Omega_0\vert$ today. 388