The bread they made on the way out of eygpt did not taste the same as what is made now. It was most likely still soft as opposed to the hard bread of today as the ovens they had were much lower temperature.
What makes modern matzah hard is the flour/water ratio. When more water is added, the bread is also softer. The issue with soft matzah is that it spoils quickly, and if you need bread for 7 days you need to make matzah during Passover.
Okay, you say, so make some matzah on Passover, right? But matzah is tricky, because if enough time passes between mixing the water and flour, it is considered leavened. (This is a religious definition, not a physical one.) Merely owning leavened bread is prohibited, so European Jews made dry matzah to prevent unfortunate slip-ups.
This custom started within the last few hundred years (again, I don't have sources, so I'm not sure about exact dates). Yemenite Jews still use soft matzah.
Edit: part of the Passover story mentions Hillel, who would wrap matzah around his Passover offering and bitter herbs. This definitely indicates soft matzah, but is much later than the Exodus itself.
there's a very early christian debate (I think settled at an early church council) if the bread of the eucharist (which many associate with matza) could be soft, or had to be hard like a cracker. So, either they were trying to distinguish it from matza (which doesn't seem to be the case), or there was crispy matza in the 1500-2000 year ago range.
Personally, I find soft matza to be terrible. our appreciation of food comes from both flavor and texture. Matza being just flour and water (no salt in traditional doughs), means it has terrible flavor (bread really really needs salt to taste not bad). Soft matza has no texture, while at least the crispy matza does.
so, a) I don't think crispy matza is a really a modern invention (they had analogues a long time ago) and b) there's a reason we moved en masse to crispy matza, its just more enjoyable overall.
Basically in the past matzo was made the day it was used because it would spoil otherwise (like any bread), but making matzo came with the risk of creating leaven which was a serious prohibition on Pesach itself.
So it's better to make it in advance, before Pesach, so they switched to dry/hard matzo that can keep forever.
These days there are freezers, and some groups make small quantities of the matzo the old way, and then freeze. It's not easy to get.