NAH

I get where OP is coming from. Basically, her partner's money builds equity and hers doesn't even though it sounds like, proportional to the income they bring in, they pay roughly the same toward running the household.

How OP phrased it is pretty awkward but when you look at it like that, I completely understand how the situation is far more ideal for him than it is for OP. She's thinking about what's financially prudent while everyone else in this thread is all, "oh but you basically get free rent!" which is way more shortsighted than what OP is talking about.

Because he doesn't want her on the title, if she cares about building equity, she'd have to get her own property. Unless she wants to blow up the relationship, she's completely locked out of building home equity if she stays with this guy and that is NOT something small or trivial (unless he's willing to change his mind if they get married).

That said, I get why OP's SO also doesn't want someone on the title if he's not married to them.