Insurance and suretyship both collect a premium, but they treat a loss very differently. An insurer expects some losses from its pool and keeps a covered loss for its own account; it does not bill its own insured, although it may pursue a responsible third party through subrogation. A surety expects no loss at all. It lends its credit to the principal and expects the principal to perform, or to pay it back.
Reimbursement: under California Civil Code §2847, when a surety satisfies the principal obligation, with or without legal proceedings, the principal is bound to reimburse what the surety disbursed, including necessary costs and expenses. Example: a surety pays the obligee $50,000 and spends $2,000 in necessary costs, so the principal owes the surety $52,000.
The surety's other rights: under §2848, a surety that has satisfied the obligation may enforce every remedy the creditor then has against the principal, up to what it spent, and may require its co-sureties to contribute, whatever order they became sureties in. Apart from those rights, §2847 gives it no reimbursement claim against other people just because they benefited.
The bond amount caps the surety, not the principal. For a bond given as security under a California statute, Code of Civil Procedure §996.470(a) limits the surety's aggregate liability to all persons for all breaches of the bond's condition to the amount of the bond. Unless a statute says otherwise, the principal's liability is not limited to the bond amount.
That is why surety underwriting looks like a credit decision: the underwriter studies the principal's finances, experience and character, because the surety is betting on this principal's ability and willingness to perform and to repay. Insurance underwriting instead relies on spreading chance losses across a large pool.