Even when a rollover is taxable, the 60 day window still applies. What happens after the 60 days is that the funds cannot be deposited into an IRA (Roth or Traditional) and any contributions made after the window of time expires would be taxable and would be required to be removed from the account or subject to a penalty for excess contributions. This penalty is 6% of the excess amount for each year that the funds remain in the account.
Taxpayers are also limited to one rollover per year.
Periodic rollover deposits aren't possible. Periodic trustee to trustee transfers may be. These are sometimes seen taken from annuities that have restrictions on when funds can be accessed - but as you noted in your research regular, periodic payments (such as those from a typical pension plan received after retirement) are not eligible to be rolled over.
It is possible for funds to be deposited into a variety of accounts, including other IRA accounts, but that doesn't mean that the deposits are allowable under tax rules. An employer might make deposits into an IRA account if a retiree directs them to do so, but they would almost certainly be considered contributions to the account, not rollovers. To make the contributions supporting earned income is required.
The simple act of a bank or other financial institution accepting funds doesn't mean that the transaction is allowed under tax regulations. The financial institutions do not have the necessary information to determine when they are or are not, so they accept them and leave it to the taxpayer to sort out later. Unfortunately, mistakes surrounding rollovers by banks and other financial institutions are quite common.
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