The sequence "[animate subject] avoir [definite animate direct object]" is meaningful in French and derives from longer sentences, as in English:

1A. J’ai Annie (à mes côtés) (I have Annie bij my side)

1B. J’ai les enfants (dans la voiture), on est en route (I’ve got the kids in the car, we’re on our way)

However, pronominalising the above sentences is a bit problematic. The full construction sound alright:

2A. Je l’ai à mes côtés

2B. Je les ai dans la voiture

The bare verb phrases sound abnormal (In the example sentences below interrogation marks (??) indicate borderline grammaticality, while a asterisk indicate agrammaticality):

3A. ?? Je l’ai.

3B. ?? Je les ai.

In particular, they sound better suited to inanimate objects, as if translating to "I possess her/them".

For third persons referents, promoting them to subject position seems to me to be the preferred strategy:

4A. Elle est là (à mes côtes)

4A. Ils sont là (dans la voiture)

A similar problem arise with the second person, made worse by the homophony of "je t’ai" and "jeter". Here, object doubling can also be used (with the third person referent, it would have created too strong of a contrastive focus):

5A. Ça ira tant que je t’ai à mes côtés

5B. *Ça ira tant que je t’ai

5C. Ça ira tant que tu es là

5D. Ça ira tant que je t’ai toi

(The absence of comma in 5D is deliberate, I don’t detect any prosodic break in such a sentence)

The first person works the same way:

6A. Tu m’as à ta disposition

6B. *Tu m’as

6C. Tu m’as moi

A similar restriction in the use of "avoir quelqu’un" in the sense of playing a trick on them; where the bare verb phrase cannot be used in some tenses:

?? Je t’avais (made better with the addition of a restrictive time complement:)

Je t’avais régulièrement à l’époque

Je t’avais eu

Je t’ai eu

*Je t’ai

Je viens de t’avoir

Je t’aurai

Je t’aurais

Je vais t’avoir

Je t’aurais eu

*Je m’imagine mal que tu m’aies

Je m’imagine mal que tu m’aies un jour.

As a beginning of an explanation, I’d posit some reluctance on the part of the speakers to put the most used of the tense auxiliaries, almost always unstressed normally, in a sentence final position where they’d receive sentential stress.

Any sentence where the auxiliary comes last can be improved by putting any pronoun, complement or dummy discourse marker after the auxiliary for it to procliticize to. This also explains why the compound tenses are unaffected by the constraint, as it’s the participle on which the stress falls in their case.

This isn’t the only context where the auxiliaries shows such a behaviour. Compare the cringey "Quel âge ai-je ?" and the perfectly acceptable "Quel âge ai-je donc ?"

"Je t’ai", while sounding weird, is grammatically correct. The reason why you never see it written without more complements is because the meaning is very vague. However, the following statements are commonly used:

Je t’ai à mes côtés.

Je n’ai pas encore tout perdu, je t’ai toi.

Je t’ai dans la peau.