Firstly, ignore the advice about complete sentences. The absence or presence of an adverbial such as a past time expression has no bearing at all on whether a sentence is considered syntactically complete.
There are occasions, however, when the listener might feel that a statement is incomplete in terms of the information it conveys. For example, if I call you and say: I had an accident, you would most probably want to know when (where, how, etc). Your statement may be contravening Grice's Maxim of Quantity, but certainly not flouting any rule of grammar.
Many statements about the past do not require an explicit past time reference - either because the time reference is clear and implicit or because an explicit time reference is unimportant.
For example, in a conversation about a trip to New York last week, you might say "I bought a new phone in Macy's." It's implicit that the buying happened last week, but the day and time are irrelevant.
So, in summary, it is not true that "a past time expression is always needed when using the simple past". And I suspect that a very large majority of sentences with a past simple form do not contain such an expression.