Obviously, your source is misleading. I do not say wrong, but “circumstances alter cases.”
Your source is talking about a gerund, which is a participle used as a noun.
The governor acknowledged pardoning his former wife, but …
I agree that
The governor acknowledged to pardon his wife, but …
is not idiomatic.
Present participles are tricky. They can be used as the lexically significant part of a compound verb, as an adjective, or as a noun.
Your source was focused on only the use of a participle as a stand-alone gerund
The female governor barely acknowledged him as lieutenant governor
can also be rendered idiomatically as
The female governor barely acknowledged him to be lieutenant governor
The female governor barely acknowledged him being lieutenant governor
The female governor barely acknowledged his being lieutenant governor
The point is that the object of the verb is now a noun phrase introduced by a pronoun, not a bare gerund. Indeed, in the second example, “being” is not even a gerund because the pronoun is not possessive.
You got a piece of advice that is correct in the narrow case of a gerund acting as the introduction of a noun phrase or a gerund standing alone, but is incorrect more generally.
EDIT In response to the comment below by the original poster, one common way to define participle is
a form of a verb used as an adjective.
This definition implies that there is no name to analyze “cooking” grammatically in this sentence.
He was cooking supper when she got home.
Clearly “cooking” in this example is not an adjective.
A common way to define “gerund” is
a form of a verb used as a noun
These common definitions may be perfect definitions in linguistics generally, but they are confusing when applied to English grammar because the same form based on a verb may be used as any one of three parts of speech. Is the word “cooking” to be designated as the nameless part of a compound verb or as a participle or as a gerund? You cannot tell if you define a participle as a form derived from a verb. Here is a definition that applies to English.
In modern English, a “participle” means one of two inflected forms of a verb, one designated “present” and the other designated as “perfect” or “passive.” Each form may be used as part of a compound verb or as an adjective. In addition, the present participle may be used as a noun. When used as a noun, a participle is also known as a “gerund.”
In short, it is more useful to define a participle in English as an inflected form of a verb that can be used as multiple parts of speech rather than as an adjective. Separate form and function.
I said at the beginning of my answer that your source was misleading rather than wrong. I’d add that the definition of “participle” that you are using is not as helpful as it could be.