There is a great deal of disagreement over good statistical style here, and indeed most of elsewhere.
But this strikes me as a mishmash of quite different procedures.
No tests for differing variances will work as designed if you Winsorize the data first. Perhaps someone has worked on this -- you might find literature references with modified tests -- but otherwise you are using a combination procedure with unknown properties. This is like doping a horse or a cyclist with something that boosts speed; you can't change the performance and be clear how much difference was yielded by the dope. You can't Winsorize and expect the tests to perform about the same. In your case, Winsorizing 5% in each tail is major surgery!
I can't speak for any statistical people but myself but outlier removal because extreme points are awkward strikes me as very poor practice.
More generally, it is now 60 years since the recently departed George Box showed that these preliminary tests are more fragile than tests comparing means, which is presumably is your main focus. I doubt I am the only one who prefers a more informal approach.
Plot the data and consider summary statistics.
If variance appears very different, considering working on a transformed scale. Logs or roots often improve the approximation to conditional normal distributions too.
Proceed to ANOVA, or if desired a generalised linear model with appropriate link function.
Apply some sensitivity analysis, e.g. ANOVA on raw data and on transformed data, to see how much difference that makes. Set aside the idea that there is one correct analysis to be identified which some oracle will reveal.