Stats used to be broken in 1.18 but were revamped in 1.19. Many of the abilities and stats have been, again, altered in 1.20; however they are far from all "broken" as many on this forum assume.

The above poster Deli has graciously posted his data on the effect of MND and Healing Magic Skill on Cure/Cura. The tentative result is that if you were to place a +15 healer hand IV in your weapon and hat (+30 total), you would raise your Cura from about 1050 average to about 1150 average. This comes out to a rough 10% increase in your potency/efficiency. If you were to double socket some pieces, this effect would be even more noticeable.

The reason the OP has issues with his "eyeball testing" is that there is inherent variation in the amount of HP you cure in this game. The variation is roughly +/-5% of the true average, meaning your values can have quite a wide spread. If you consider the above example with 10% increase in potency, if you only cured yourself 2-3x in each set you may not even notice a difference. In addition, when the OP mentions doing 1500+ on a Cura, he's clearly spiked a "Critical" hit, it just doesn't appear in the log. These things need to be taken into account.

Beyond this, INT makes a huge difference on the damage of spells on higher level mobs. Alternatively MATK makes a huge difference on mobs at lower levels. These things are so obvious that you could even eyeball test them and notice a difference.

If you've gotten this far in the post and truly care about how stats work in this game, I suggest visiting BG forums. There is an ongoing stat testing thread there that may be of interest to some. While there is some good information on official forums, game mechanics and testing are not this forum's strong point. It is true some stats still don't "matter", but most do now. The community in XIV is very weak (due to the game essentially flopping and now currently attempting a revival under YoshiP). As a result, you will not find these formulas plastered on the main forums or on a wiki, but they do exist for those who care enough to check non-official forums.