While I certainly agree with the comments that cure wounds should have a bit more "omph" to it (maybe make it 1d12 + mod or add caster mod to every level of spell? Or at the very least make it 30ft range instead of touch) to help it compete with healing word, I have a fundamental issue with your the proposed changes the OP is proposing. I don't like the idea of forcing the players to spend healing dice in order to gain healing from a spellcaster.
First, it defeats the entire purpose of giving players short rest dice, to help make it so the cleric/druid doesn't have to spend ALL his/her spell slots each day to keep the party up and active. The monsters in 5e tend to get very swingy, especially at higher levels and the game expects players to have access to these dice to help offload the healing. With forcing to party to spend these dice when the cleric casts a heal on them you are effectively double punishing the cleric, who already has already chosen to spend his spell slots on healing instead of something interesting (whether combat healing is effective is irrelevant in this regard), and by reducing the rest of the party's self healing you make it more likely they will need more spell slots dedicated towards heals than normal.
Second, giving players short rest healing is to incentive every class to be willing to take short rests. Many classes regain features when the party short rests and spending healing dice to heal is another carrot to convince barbarian and rogue that maybe they are better off if they let the warlock take a quick breather rather than kick in the next door in search of heads to bash and loot to steal. Players short resting can even be good for the DM, as it gives intelligent monsters time to place traps or make plans if the party decides to take a nap in their foyer. This change drastically alters the amount of healing for the party and therefore the pace of the game.
Third, this has a larger impact than just player characters. If you change the cure spells, how do you handle NPC spellcasters? Do they use these same rules in combat? What about out of combat, like if they hire a priest to heal them? If they function differently than the player characters, why? With commoners or more accurately npcs who can afford magical healing from other npcs, why would they bother to seek out a priest from a mechanical standpoint (the class in trope of a group of farmers bringing an injured child to a church and begging for care or pooling their wages) when they could short rest for roughly the same amount? What about healing potions? Or the healer feat? I could keep going. It brings up a LOT of potential questions...
