Can a school that only provides proof of school payment when going to the UK apply for a visa?

In your case, it may be risky to provide only the proof of payment and the letter of guarantee issued by the school. The application materials for a British visa need to prove that you have enough funds to pay all the living and study expenses during your stay in the UK. Although you have provided a letter of sponsorship for the school of 30,000 pounds, it is still recommended to submit a personal bank account or other financial resources.

Generally speaking, in the short-term exchange program in the UK, if your bank transaction records in the three months before entry can prove that you have enough funds to pay the whole travel expenses and maintain your life in the UK, then you only need to submit the bank statements, work income certificates and other relevant materials in the last three months.

For specific requirements, please refer to the list of materials required for Tier 4 ordinary student visa in official website https://www.gov.uk/Tier-4-general-visa/documents-you-must-provide. In order to better ensure your visa success rate, it is recommended to consult a local professional immigration lawyer or consulate to verify and standardize the operation.