why do we see characteristic colors of alkali metals in oxidizing flame?
I don't think it is a good idea to connect ionization energy with the color of flame emission, especially in a Bunsen burner.
Ionization energy means that you have separated the electron out of the nucleus attractive field. However, electronic transitions, corresponding to visible wavelengths, take place while the electron is still bound to the nucleus. Thus the flame color of alkali metals might correlated with the IP  but not the causation.
Let us look at the resonance wavelengths of emission of Li, Na, K, Rb, and Cs. Resonance wavelengths means the transition from the $\mathrm{1s}$ to $\mathrm{2p}$ orbital or $n\mathrm s$ to $(n+1)\mathrm p$ orbital in the case of alkali metals.
$$
\begin{array}{r|ccccc}
\hline
\text{Element} &amp; \ce{Li} &amp; \ce{Na} &amp; \ce{K} &amp; \ce{Rb} &amp; \ce{Cs} \\
\hline
\lambda/\pu{nm} &amp; 670 &amp; 589 &amp; 769 &amp; 794 &amp; 894 \\
\hline
\end{array}
$$
You cannot even see these wavelengths with your eyes after 700 nm which are in the deep deep red. Some people can see the red lines of K 769 nm. However, you can see the energy gap is indeed decreasing as we go down the group. The ionization energy also goes down the group. However, you can see, the colors which you see in the Bunsen burner have nothing to do with ionization energy.
The wavelengths have been extracted from a old book titled Graphische Darstellung der Spektren von Atomen und Ionen mit ein, zwei und drei Valenzelektronen by Dr. W. Grotrian, 1928. I think NIST atomic spectra database will be better today but I am not sure if they list resonance lines clearly. This book has graphically labelled transitions for almost all elements known in 1928.