In the Ramsey model, how does it make sense to say that capital equals wealth when wealth can be negative and capital cannot?
No matter in Ramsey model or the basic structure of Solow-Swan model, as long as is in the closed economy. The asset equals to the capital in the equilibrium. 

$A_t$ is the total asset owned by the households, or by the representative household. In the market equilibrium, all the borrowing and lending must cancel within the economy. Therefore, the only asset in positive net supply is capital, which means $A_t = K_t $. The asset equals the capital stock quantity.

Therefore, the asset per person $a_t$ = the capital stock per person $k_t$. It seems like you confuse $A_t$ with $a_t $.