In a patient with an abscess on the face, should the tooth be opened for root canal treatment, or should antibiotics be given first and treatment started after it has improved?

If mouth opening is sufficient and the patient is not in a very poor systemic condition, our first choice is to open the tooth, establish drainage and support the procedure with antibiotics. If antibiotics alone are given without opening the tooth, the abscess does not drain sufficiently and the patient suffers considerably until they take effect. Antibiotics encapsulate the abscess, making it harder to resolve. You can look at the relevant case at the link.
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