What you say is already done extensively by the various intel/and/etc.
Of course there a full test such as hard drives is not feasible since the structure is far too complex and a full test would take far to much time. Test and characterisation is however done at a higher level.
For example let’s take intel lineup for familiarity, it’s well known that low to mid end Xeon, i9 and I7 use the same silicon die. They all start out as Xeon, the ECC memory and busses for dual cpu support are tested, if any of the two don’t work the die is downgraded to I9, now the cores are tested, if some of them aren’t working correctly the die is downgraded to i7 etc
Also another thing to note is that while we think to them as such mosfets are not digital switches, they are analog transistors, so it is quite unlikely that one isn’t switching at all, a far more likely defect is a transistor that cant switch at the maximum designed frequency.
That is why in intel lineup there are dozens of SKU’s with marginally different frequency, so that defective xeons aren’t thrown away, just sold as low end I7’s for less profit
EDIT: also error correction in general adds a lots of overhead so in order to add ecc logic to the whole design you might need to add 20 to 50% more transistors to the whole CPU, that could otherwise be put to much better use.