Witches are natural magic users
This may seem like a weird reason, as it would make sense that someone who's a natural without tools can be better enabled by tools. Give a math genius a calculator, you enhance their abilities. Give a runner a good pair of shoes, you enhance their abilities. Et cetera.
However, what if witches were trained to do all that without tools? They've been taught how to do all the runes and rituals at lightning speed, and as such even placing a wand in their hand provides a great mental block. A true math genius doesn't need a calculator (or much time) to do all but the most difficult calculations (I remember on the show Stan Lee's Superhumans they found a guy who really excelled at mental math, to the point that professional mathematicians calculator input speed was slower than his mental speed. In that case, the extra device would be useless to him).
Furthering the previous point, witches were proud magicians when wands were first introduced, meaning they refused to touch them because only "those people' needed them for good magic. If you needed a wand, you were no better than "those people," who were so weak in mana they can't cast spells on their own.
This kind of doctrine leads to a kind of social incest, where witches hang around other witches and only reinforce their beliefs, leading to wand makers to be seen as saints/missionaries that give those poor unfortunate souls the table scraps of witch society. Eventually, if enough people believe that it's impossible, then if witches are taught that way in school, they will eventually not be able to believe that they can use wands. Just think about what the US teaches about Soviet contributions during WWII.
I find that if I can't explain something logically, then "tradition" or "taboo" (the negative form of tradition, ie "do/don't do this") is probably the most satisfying answer.
Magic from a wand isn't the same
As much as a wand can hold the spell inside of it, there's a missing component that keeps it from being as effective as real magic. For example, in The Irregular at Magic High School, which has a similar dichotomy between Old and New magic, it's stated that old magic has extra rituals in it to provide extra effect, such as providing stealth casting during already long rituals. Perhaps wands lack some element that makes the spell more effective, meaning that witches, already naturally being able to do better wands can do, would go without (again, wands are witch table scraps).
Witches are magic specialists
Witches act as magic specialists that use their high level of expertise to help with a wide variety of problems using magic. A housing contractor doesn't have all the plumbing, framing, electrical, HVAC, windows, foundations, and landscaping equipment, as they simply can't carry that around to every job. Well, your witches can't even carry around all the spells they need for regular usage on magic sticks, plus they can break, get worn down, etc, and then they need to get some more, which is a loss for them due to lost opportunity. Witches need their complex knowledge to turn a profit, doing complicated runes and rituals for specific spells.
Spells look cooler
Your witches have an image problem. Muggles are scared of witches suddenly blasting away everyone with a wand spell, so Witch Council or whatever banned the use of wands. You now have witches looking cooler doing normal spells to be more entertaining rather than threatening.