I've been trying to bake bread using homemade flax eggs, but every time I do, the inside of the bread is mushy and uncooked. What am I doing incorrectly?
It may be that eggs serve a different purpose in the recipes you experimented with.
Eggs can be either a leavening agent, emulsifier, thickener, binder, coat and sealer, or color and shine. (source)
From the recipes you posted, it looks like one was more of a leavening need (90 second recipe) and the other was more for moistness (sandwich bread recipe).
Gluten free bread calls for eggs because...
Eggs are natural leaveners that help boost the rise and volume of bread. Eggs also add moisture, flavor, and protein to gluten-free bread recipes. If you select a gluten-free bread recipe that includes eggs, you have a better chance that the resulting bread will have good color, more volume, and softer texture. Eggs also help to extend the shelf-life of gluten-free bread, which can become stale more quickly than wheat bread. https://www.thespruceeats.com/tips-for-better-gluten-free-bread-1451069
Flax eggs appear to be best used for binding because they don't offer much structural support (source).