Well, getting elements from the template turned out to be not that strightforward. Since it's a custom style, it should be applied to every element there is being extracted:
var style = _assetLoader.Load<StyleSheet>("UI/Views/SteamUpdateButtons");
var template = _visualElementLoader.LoadVisualElement("SteamUpdateButtonsTemplate");
var button = template.ElementAt(0) as Button;
button.styleSheets.Add(style);
var image1 = template.ElementAt(1);
image1.styleSheets.Add(style);
var image2 = template.ElementAt(2);
image2.styleSheets.Add(style);
Not exactly simpler than making each control separately and adding stye/class.