What herbs are you talking about? some will do fine outside, by dying back for the winter.
Most of the mediterranean type of culinary herbs wouldn't notice the cheap lighting. A good southern window exposure might be stronger than a basic grow light from a home repair store. Those are great for seedlings but don't do much for plants that require full sun.

Most of them are the basics. Thyme, Basil, Oregano, Lavender, spearmint, etc. So I shouldn't really worry about super good light?

Different plants have different needs. Thyme, oregano, sage, tarragon, mint, lavender are perennial -- leave them outside. Basil is an annual, plant new ones each spring. Rosemary is a tender perennial; in zone 5 it needs to come in for the winter.

The hardy perennials need a cold winter and will die within a year or so if overwintered indoors. Not to mention the root shock of being dug up and potted. Lavender is also tender.

Most herbs seem to benefit from being mulched for the winter.