Does the use of a guarantor give the tenant an automatic legal
responsibility to repay them?
Yes, unless a contract expressly says otherwise.
A guarantor has the right to recover what the guarantor had to pay from the primary person obligated on a debt that are the same as the creditor who the guarantor pays unless a contract or statute says otherwise. This right is called the right to subrogation.
The subrogation rights of a guarantor are a matter of equitable principles set forth in case law precedents (and in the case of promissory note, a statutory right under the Uniform Commercial Code, see, e.g., Colo. Rev. Statutes § 4-3-419(e)), even though best practice is to formally state that in a guarantee agreement. See, e.g., Behlen Mfg. Co. v. First Nat. Bank of Englewood, 28 Colo.App. 300, 472 P.2d 703 (1970). ("‘It is well settled that where one secondarily liable is called on to make good on his obligation and pays the debt, he steps into the shoes of the former creditor. He becomes subrogated to all the rights of the creditor against the principal debtor, including the security given to secure the debt.’").
The law as stated in an insurance context is that:
When an insurer reimburses a victim for damages pursuant to a claim
under the victim's insurance policy, the insurer enjoys a right to
subrogation, under which he can stand in the victim's shoes and
collect the reimbursed amount from the party responsible for the
damages. Am. Family Mut. Ins. Co. v. DeWitt, 218 P.3d 318, 323
(Colo.2009). The right can arise pursuant to an express provision in
the insurance policy—a “conventional” subrogation right—or under
principals of equity—an “equitable” subrogation right. Id.
Ferrellgas, Inc. v. Yeiser, 247 P.3d 1022, 1027 (Colo. 2011).
But, the law of subrogation of guarantors is the same as that of insurers, who are both a subset of people who are called "sureties."
Colorado's common law right is referenced obliquely in its statute governing contribution rights (see below) for sureties:
Nothing in sections 13-50-102 to 13-50-104 affects or changes the
right of a surety who has paid his proportionate share of an
indebtedness of recovering the same from his principal debtor.
Colo. Rev. Stat. Ann. § 13-50-104.
Similarly, in the case of a release of some but not all joint debtors or guarantors, Colorado has a (widely adopted model) statute which states more affirmatively:
In case one or more joint debtors are released, no one of the
remaining debtors shall be liable for more than his proportionate
share of the indebtedness, unless he is the principal debtor and the
debtor released was his surety, in which case the principal debtor is
liable for the whole of the remainder of the indebtedness.
Colo. Rev. Stat. Ann. § 13-50-103.
When there are multiple debtors, or multiple guarantors, the obligation is generally joint and several, so that the entire debt may be collected by the creditor from any one of them, not just their fair share of the debt.
But, there is also a common law and statutory right for guarantors who pay more than their fair share of a guarantee obligation upon which there are multiple guarantors to recover the fair share of any guarantor who did not pay their fair share of the guaranteed debt from that other guarantor in an action for contribution.
It is well established that guarantors paying more than their
proportionate share of an obligation are entitled to contribution from
other guarantors who are jointly and severally liable for the
obligation. Taylor v. Hake, 92 Colo. 330, 20 P.2d 546[.]
Stieben v. Korby, 533 P.2d 530, 531 (Colo. App. 1975).
Incidentally, this is an area of law upon which United States law is generally very uniform, even though it is, as most things are in U.S. law, a matter of state law and not a matter of federal law, in most cases.