There is some evidence of initial *h₁ in roots like *h₁es- "be". Alex B in the comments quotes an exercise from Fortson's textbook, Vedic ā́sat "monster" from *n̥-h₁s-. If the root were **∅es-, we would expect **n̥-∅s- > **ás- with a short vowel. But reconstructing it with the laryngeal, *n̥-h₁s- > ā́s- due to compensatory lengthening.
Arnaud Fournet mentions some additional evidence from Tocharian, and Kloekhorst suggests that Anatolian shows an initial ʔ- as a direct reflex of *h₁-, but this is rejected by most other Anatolianists (like Melchert).
Still, though, there are a lot of roots reconstructed with initial *h₁- where there's no direct evidence from any of these sources. Regarding these roots, Alex B gives another quote from Fortson:
Specialists differ on whether to rewrite all traditionally vowel-initial roots in this way when direct evidence of a laryngeal is lacking; the tendency is to add the laryngeal regardless, for the sake of structural uniformity.
In other words, it's elegant to have all roots consist of two groups of consonants, so *h₁- is used when there's no evidence of any other consonant.