Hey Cliffy. The answers to your questions are quite simple and obvious, so I was reluctant to spend time pressing keys on my keyboard to reply.
Basically, if AMD had their owns fabs, they could manufacture many more CPU's. This would allow them to gain market share vs Intel. Last quarter, they actually lost market share, as reported by the BBC in the opening post.
Intel have 15 fabs of their own, which allows them to produce millions of CPU's more than AMD each year.
In regards to your other questions:
A) Why Intel have been stuck for so long on 14nm when having their own FAB is such a benefit
They've been stuck on 14nm as developing 10nm or 7nm is very difficult.
B) why Intel themselves have struggled with shortages in chip production over the last few years
There's huge worldwide demand for CPU's, which will only get worse over time. Intel can also sell every CPU they produce, despite having huge manufacturing capacity. Intel actually spent billions doubling their manufacturing in the last 3 years.
Intel announced a few months ago that they'll be purchasing additional wafers from TSMC, this will allow them to produce even more CPU's, so will likely further increase their marketshare. In addition, TSMC currently have a more advanced process than Intel, so the Intel CPU's produced by TSMC will allow Intel huge performance gains.
This is bad news for AMD, as it'll increase the price TSMC charge per wafer and will mean less wafers/CPU's for AMD, which will likely cause it to lose further market share.