Was [this] a native word that ... just took on the kanji?
In short, yes. :)
The longer details
For any word in Japanese, you have to ask first, "is this a kun'yomi native Japanese term, a 大和言葉【やまとことば】? Or is this an on'yomi borrowing from Middle Chinese, a 漢語【かんご】? Or is it some combination?"
These three questions drill down to the history of the word. The oldest Japanese words are (generally speaking) the kun'yomi. Borrowings from Middle Chinese were brought over mostly in the 400s through around the 700s. Combinations were later coinages in Japan, mixing and matching pieces from these earlier two boxes.
The word hikaeru is a native Japanese term. Consequently, we can ignore the kanji: pretty much all kanji spellings for kun'yomi words are after-the-fact applications of a borrowed non-native writing system, and thus the kanji spellings are not fully relevant to the origins of the term's derivations and core meanings.
General derivational patterns
There are many verbs in modern Japanese that end in -aeru. Many (but not all) of these verbs originally derive from a base verb in the 未然形【みぜんけい】 ("irrealis or incomplete form", for type 1 verbs this is the stem ending in -a) + auxiliary verb ending ふ denoting repeated or continued action, or ongoing state.
- This auxiliary had the 下二段活用【しもにだんかつよう】 ("lower bigrade conjugation") pattern, where the different conjugations ended in either -u or -e. These pretty much all simplified over time into just the -e stem, with the terminal (sentence-ending) and attributive (noun-modifying) forms ending in -eru.
- Many medial (mid-word) "f" sounds softened further to "w" and then vanished.
- → These two historical trends together mean that ancient auxiliary ふ later became へる, and then modern える.
So the key to understanding the many uses of these verbs is to look at the base form.
Specifics of 控【ひか】える
For 控【ひか】える, more detailed monolingual Japanese dictionaries show that this has the older kana spelling ひかへる, and derives from the older form ひかふ. This tells us that we're dealing with 未然形【みぜんけい】 + auxiliary ふ, so we can find the base form by removing the -aeru from the modern verb, which here leaves us with hik-. This is the root of modern verb 引【ひ】く, "to pull, to draw close". The auxiliary adds the shade of meaning of either "to be pulling, to be drawing close" (as a repeated or continuous action), or "to keep close: to have pulled, to have drawn close" (as an ongoing state as the result of the action).