As a vegan baking enthusiast I am also interested in this problem.
It looks like the kind of bread you're talking about is an enriched bread - those are the ones that typically have the ratio of ingredients you describe (a high proportion of tenderizing ingredients like sugar, butter, and eggs, as opposed to leaner doughs).
While it is true that some traditional recipes are hard to make without eggs, there are still many bread recipes that can be made by either substituting or omitting them.
In some cases it's just a matter of increasing or adding chemical leaveners and some acid in the batter (certain cake recipes, for example), in others, it just requires adding more moisture (sometimes silken tofu is used, vegan yogurt, or mashed banana). For binding, ground flax seeds in some water turn into a gel, and then there's aquafaba, which has emulsifying, foaming, binding, gelatinizing and thickening properties, and can be used in place of eggs and even whipped up into a meringue!
In enriched breads, I think you'll stand a good chance with substitutions since eggs don't play such a critical role in terms of leavening or binding. The eggs are probably there to add richness, flavour, and as already suggested, to create a fine soft crumb.
Assuming this is the case, I have several suggestions (which I'm currently in the process of testing out myself!):
- adding potato starch or cooked potato which will make them more airy and fluffy (potatoes soak up more moisture than wheat flour, this is why some recipes use bread flour instead of AP for enriched dough, because you can use more tenderizing or liquid ingredients becausue bread flour absorbs more moisture. I've seen potato used in cinnamon roll and donut recipes too). Another reason is that potato contains starch but no gluten, and gluten can make baked goods tough.
- Using 'tangzhong' method: cooking some of the flour and water ingredients on the stove top till it almost looks like pudding. Doing so can result in a very moist, feathery, airy crumb. ATK has a good article on this.
Those are the two that come to the top of my head. What I have done in the past is either just left the 'egg' element out of the enriched dough completely, or compensated by adding a little more fat or liquid.