Zinc isn't the problem. It's the phosphorus in the ZDP that clogs up the cat.
Basicly, the zinc was removed to protect the emission equipment, but other additives have taken it's place. So, except for possibly breaking in a rebuilt engine, the ZDP isn't needed. In regards to emission testing, it itself won't show up, but you might be left wondering why you need a new cat before 150k miles.
ZDP was first added to engine oil to control copper/lead bearing corrosion. Oils with a phosphorus level in the 0.03% range passed a corrosion test introduced in 1942.
In the mid-1950s, when the use of high-lift camshafts increased the potential for scuffing and wear, the phosphorus level contributed by ZDP was increased to the 0.08% range.
In addition, the industry developed a battery of oil tests (called sequences), two of which were valve-train scuffing and wear tests.
A higher level of ZDP was good for flat-tappet valve-train scuffing and wear, but it turned out that more was not better. Although break-in scuffing was reduced by using more phosphorus, longer-term wear increased when phosphorus rose above 0.14%. And, at about 0.20% phosphorus, the ZDP started attacking the grain boundaries in the iron, resulting in camshaft spalling.
By the 1970s, increased antioxidancy was needed to protect the oil in high-load engines, which otherwise could thicken to a point where the engine could no longer pump it. Because ZDP was an inexpensive and effective antioxidant, it was used to place the phosphorus level in the 0.10% range.
However, phosphorus is a poison for exhaust catalysts. So, ZDP levels have been reduced over the last 10-15 years. It's now down to a maximum of 0.08% for Starburst oils. This was supported by the introduction of modern ashless antioxidants that contain no phosphorus.