Levene test for equal variance
&quot;As I understand, if the p-value is smaller than 0.05, then I can reject the null hypothesis and tell that the variances are not equal (and then it makes sense to use ANOVA.&quot;
Your understanding is incorrect. As a measure of goodness of fitness of data, null hypothesis has to be accepted if alpha is less than  or = .05. If you want to perform statistical test of variability across groups, you need  an implementation of Chi-square test of variability across groups.
Your intuition (question in last part of body) -p-value should be greater than 0.05  is not logical. This value indicates sampling fluctuations or error arising on account of chance in sample selection. An (sampling ) error greater than 5% is not acceptable by any standards.