This looks like a Privet (Ligustrum), not a holly. (Notice the leaves are opposite each other. Hollies have leaves that alternate along the stem.) A couple of privet cultivars are golden-leaved, so maybe the original plant was in too much shade and looked greener at the time.

Rosemary often get spider mites. Cutting it way back and letting clean foliage grow out is a good option.  Yes, you can spray with insecticidal soap or horticultural oil and still safely eat the rosemary. Read the labels carefully. Both can burn.https://extension.umd.edu/resource/spider-mites-vegetables

Ellen

It's likely the older plant is a Privet as well, or perhaps a volunteer weedy Privet was growing intermingled with the holly. (They are highly invasive, and a bird eating holly berries or roosting in the tree could certainly have dropped Privet seeds at its base.) You can always send in a picture of the original plant for us to try to ID if you wish.

Privet are quite pH-adaptable, so a fertilizer like Holly-tone for acid-loving plants is not needed, though not harmful. Early spring is a bit early for application of an organic fertilizer like this since soil microbes break it down for plant use, and they aren't very active yet in cool soils. Still, it is a slow-release type and should last the plant until fall or even next year.

Unless the golden color is the Privet's natural appearance (in enough sun), off-color leaves usually indicate an issue with root health. Since Privet are so tolerant of a range of growing conditions, the most likely factor that causes them stress is overly-wet or poorly-drained soil. Nutrient deficiency is possible, but not likely given that it usually affects only a portion of the foliage (youngest or oldest leaves first, depending on the nutrient in question), and often results from poor root function rather than an actual lack of nutrient content in the soil.

The alternative to discoloration is sublethal exposure to herbicide. Was any type of weed-killer used on the lawn surrounding the plant? Some chemicals are risky to use near woody plants (trees, shrubs) because of the danger of root and bark absorption, though they will state this caution on the product label. If that happened here, all you can do is wait for the plant to grow out of the exposure if the root damage wasn't too great.

Aside from this, all you can do is to give the plant time to absorb the nutrients it may need, and to monitor the soil moisture levels for watering needs, as with any new planting. It should only need soaking in times of drought (dry weather for several weeks, for example).

Miri