Has the custodian already refused to issue you distributions? What makes an IRA different from a transfer-on-death or TOD account, is that IRA distributions must be reported to the IRS, with is not the case (necessarily) with a regular bank account or TOD brokerage account. Non-IRA accounts report only the income earned on the account, and if there is no Social Security number on file, the bank or brokerage would subject the earnings to backup withholding. Once the financial institution determines you are the correct beneficiary, it can just issue you the funds without providing you with a 1099 form at all. However, there is no backup withholding on an IRA account, the custodian simply cannot make a distribution with a Social Security number.
A beneficiary designation (BD) form might only contain your name, address, date of birth, and relationship to the account owner, and not your Social Security numbers. Even if it BD form does include your Social Security number the financial institution must ensure it has your correct number, received from you, so it can then issue you the proper 1099-R along with the proceeds of the account.
You are probably right that a financial institution does not legally have to retitle the account, however, they are within their right to insist that a new IRA account does get set up. It does do not much obligation to turn funds over to you, at least not right away, and not without you providing them with the information they may be asking for. By formally setting up new IRA accounts it can ensure that it is not complied with federal laws and regulations requiring them to record an verify information about its customers as well as ensure that the 1099s have been issued to the correct beneficiaries. When you inherit an IRA from a non-spouse, you have at least five years to take liquidate the account, and some people thinking they need to take a distribution immediately have sued the financial institutions for not advising them of all their options.