Could all CPU instructions get fundamentally faster if a better multiplication method was developed
I don't see the relevance.


New multiplication algorithms only offer a practical improvement for numbers with huge numbers of bits and probably don't beat existing algorithms until you're dealing with numbers with thousands or even millions of bits. For an extreme example, F&uuml;rer's algorithm, which runs in time $O(n\log n\, 2^{O(\log^*n)})$ will only become competitive with Sch&ouml;nhage and Strassen's $O(n\log n\log\log n)$ algorithm for $n&gt;2^{64}$ bits and, let's be honest, whatever it is that you're multiplying, it doesn't have $2^{64}$ bits. In contrast, CPUs operate on fixed-length words with a few tens of bits.
Most of what a CPU does isn't multiplying numbers, anyway.
Also, note that Harvey and van der Hoeven "make no attempt to... exhibit a practical multiplication algorithm" (page&nbsp;2 of their preprint).