"Simple present" is a reasonably good name for this construction. There is only one word in the complete verb, so it makes some sense to call it simple. I prefer to call it "present indefinite".
Many textbooks lump all the properties of a verb construction under the heading "tense". I find it easier to explain those properties separately: as voice, mode, tense and aspect. Your example sentence uses the active voice, indicative mode, present tense and indefinite aspect. In short, the "present indefinite".
Leaving voice, mode and tense as they are, there are four available aspect constructions:
Indefinite: I have four children.
Perfect: I have had four children.
Continuous: I am having four children.
Perfect continuous: ? I have been having four children.
I've marked the perfect continuous aspect as questionable. It doesn't make much sense when regarding the bearing or possession of a specific number of children, but "I have been using two accounts" is a sensible present perfect continuous construction, as is "I have been having children for as long as I've been married".
In the indefinite case, there is nothing that indicates aspect. Nothing indicates a perfect aspect. Nothing indicates a continuous aspect. There is no indication of aspect at all. From the grammar alone, we can't tell what aspect the statement has. The aspect is indefinite.
In the perfect-only case, we can tell that the aspect is perfect. The "have" (a form of "to have") is an auxiliary verb which allows (or licenses) the perfect aspect. The "had" is a so-called past participle which shows the perfect aspect (or satisfies the license).
In the continuous-only case, we can tell that the aspect is continuous. The "am" (a form of "to be") is an auxiliary verb which licenses the continuous aspect. The "having" is a so-called present participle which fulfills the continuous aspect license.
In the final case, the rules for both perfect-only and continuous-only cases are fulfilled. The "have" is a present-tense verb (a form of "to have") that licenses the perfect aspect. Following it is the so-called past participle "been", which is a past participle form of "to be" that in turn licenses the continuous aspect. Following that is the so-called present participle "having", which fulfills the continuous license. The perfect continuous satisfies the rules for both the perfect (some form of "to have" followed by a past participle) and the continuous (some form of "to be" followed by a present participle).
Your gut might be trying to tell you something different. Outside of the grammatical properties, some verbs are dynamic while others are stative. This is a semantic difference, but it is a difference which impacts grammar. Because "to have" (especially "to have children") is a stative notion, dynamic transformations don't make sense. For example, the perfect continuous doesn't make sense in the absence of supporting context.
In short, "I have four children" employs the active voice, present tense, indefinite aspect, indicative mode, and stative semantics.
There is only one marker for tense, no markers for aspect, no markers for mode, and the stative semantics come solely from the definition of the one word in the verb.
