It also doesn't allow healing to be useful when the entire party is at full health.

Fell Seal: Arbiter's Mark has the Arcane Engine ability, which converts excess healing into MP. While there's only one character who can learn it (even with the DLC), it can still be used to great effect; use a Full Potion on the character, then use that one spell that does damage that scales with the caster's MP. It allows for massive damage to be dealt once during a battle, and that can be useful for killing a boss before its HP gets low.

You've got a lot of different options for healers in Pathfinder. There is a White Mage archetype of the Arcanist class that is essentially an arcane caster with spontaneous healing abilities. A Cleric is probably the better choice for healing in general since they can remove various debuffs as well as heal damage with most archetypes. A Witch has a nice mix of offense and healing in their spell list - and you can make one focused on healing by giving them a Healing or Mercy patron.

If you want someone who can excel at both combat and healing you won't go wrong with a Paladin - healing competently does come in at later levels though, not at the start. That healing is very strong though at high levels, able to heal a large amount of health as well as remove a lot of debuffs in a single use because of Mercy effects.

Initiative is actually quite an important stat for healers, and it turns out that, in a Wizardry/Dragon Quest-like battle system, it helps for initiative to be either high or low. A couple interesting examples:
* In Stranger of Sword City Revisited, the Clocker (which is more like a Red Mage, but more frontloaded) has two abilities that allows them to cast a spell twice at the beginning of the round, at the cost of not being able to do so next round. This is, needless to say, quite handy when combined with healing magic (especially if you multiclass to combine these abilities with more powerful spells).
* In SaGa 2, many late game bosses rely on being able to constantly heal the entire party. In this case, it helps for your healer to be *slow*, so that the enemy reliably acts before the party heal that heals the damage. (Reliably acting before the enemy is not possible because of a bug, unless you're playing the DS version that fixed that bug.) (Worth noting that this is *not* typical of the SaGa series; SaGa 2 and 3 are the only two games in the series where multi-target healing is readily available, with SaGa Frontier having rare skills that do so (usually at the cost of LP), and Romancing SaGa 3 having rare staves that can heal the party with a skill called Shatter Staff, and that's usually not practical because of the drawback that the name suggests.)

How many of these options are actually implemented in the computer games?

depends completely on the game

D&D et al have the concept of "temporary hit points", which are nudges above your max. Interestingly (at least with the editions I'm familiar with), they're usually the realm of arcane casters than divine casters. Example:

https://legacy.aonprd.com/coreRulebook/spells/falseLife.html (2nd level healing spell would be 2d8+CL; this is 1d10+CL temporary HP). For game balance reasons, the games usually prevent you from having multiple sources of temporary hit points adding up. (Ie, no "cast false life 4 times before combat on one person").

There are also things like Pathfinder's Lifebound feat that allows you to keep healing over maximum as temporary hit points for a short time. https://aonprd.com/FeatDisplay.aspx?ItemName=Lifebound .

Some board games also have that too. Middara has an item a character can equip where (once per fight) extra healing over max charges the item, and then later you can spend an action to suck that life out of the item to heal from it.

These are pretty much all things that a video game could easily-enough do if it was built to, but, oddly, are rarely tried. Have you seen games with such?

The typical cleric in a DnD setting? All for it!! (one of my favs along with bard and ranger...)

I've also seen temporary HP in CRPGs. For example:
* In TES: Arena and Daggerfall, Shield spell effects work like temporary HP (and are *really* powerful in Arena; say hello to 99 HP/level, with no finite duration so it stays when you fast travel)
* Wizardry 8 has Guardian Angel
* Even Centauri Alliance has a shield mechanic; by activating certain suits of armor, you can get up to 100 points of shielding per character.
* In Dungeon Hack, the Vampiric Touch spell grants the player temporary HP with unlimited duration, and it *stacks* with itself (clearly a mis-implementation of the spell).

In Wrath of the Righteous - all the options I listed have been implemented. Kingmaker is much more limited in class choice compared to WotR - no Arcanist (or White Mage), and no Witches.

In WotR, the main character can be an amazing healer regardless of their class depending on their mythic path choice. Mythic paths are essentially a demigod class separate from your main class and they tend to outshine specialists in their respective field (eg. a fighter who becomes a Lich will always be better at necromancy than a wizard who specializes in that field and doesn't follow the Lich path - likewise a cleric who doesn't take the Angel path isn't going to be anywhere as good as rogue who becomes an Angel at healing and buffing). Ideally your main character will be a class that complements a particular mythic path (like a cleric angel).

I looked up some information on WotR, and it appears that some of the other Mythic Paths have some healing. In particular, the Trickster path has some interesting (and unusual) healing options available; not good for mass healing, but can eventually get unlimited healing with the right setup (costing only a turn to heal).

Cleric 100% for me. I respect the pure support, but I always prefer a character who can hold his weight after the buffs are up or things get nasty in the back line than a character who is purely support. Though in multiplayer that changes. I might think about this more deeply later.