The only alternative approach is that Characters fall in love with NPCs because some participant (player, GM, etc) decided it was so.
It's either dice or decide.
I've only seen a small handful of games where falling in love is forced by mechanics.
Pendragon is one. Fate allows creating temporary aspect, and that can, without too much of a stretch, be a means of imposing a love (or hate, or disdain, etc) upon another.
Vampire The Masquerade 1e is another, but it's a part of the blood bond mechanic, magic mind control.
Ars Magica, it's a Rego Mentem spell effect, or a faerie power. It's also doable in
Mage: The Ascension, but I don't remember the forces used.
ElfQuest has the Recognition mechanic - it's not love, but it's irresistible rut instinct; it's also rare, and nigh-guarantees children resulting.
Of the games I've seen where it is mechanicalized, only
Pendragon and
Fate don't make it magic... tho' Love in pendragon has some profound effects. Including, if in a romance, a noticeable annual glory boost. And having a love score can do nigh-magical things in play via the Inspiration mechanic, so it's thematic and usually a benefit...
And
Fate? Well, temporary aspects are just that - temporary, so unless the player opts to adopt it permanently, or a major concession on a social is needed, it goes away... but can be compelled a lot in the mean time.
Cortex Plus and Cortex Prime also have
I'll note that R. Talsorian Games' RPGs almost all have background generators, and those can put good and bad romances in the character's past... but no mechanic for the present. Mongoose's version of
Traveller does, too. But again, past only, not present nor future, and far less often than RTG. Jennell Jaquays' (as Paul Jaquays)
Heroes of [x] series also included past romances good and bad.
Mechanicalizing Love an area where few games explicitly go, but the ones I've read that do do so for strong setting based reasons...
But Fate, Cortex Plus/Prime, and Pendragon, any long term (or in Fate and Cortex, short term, too) emotional state can be mechanicalized within the rules frameworks, and love is just a particularly potent one.