What can't enzymes still solve or be used because of their delicacy
and instability even with the current advancements in technology?

(If enzymes were very stable like 5 months, where can they be used
that currently cannot and why? Industries, applications, products...)

magic words -  catalytic converter poisons

Basically keys that get jammed in the lock and can not be cleared out.

Perfect feed stocks equal long operation.

Technically catalytic converters do not use enzymes, which are protein
molecules. Most protein quickly become denatured at high temperature
or under adverse conditions (acid, heavy metals, etc) and are
therefore usually not useful for long-term usage. It might be possible
to find or bioengineer more resistant forms: the enzymes from the
bacteria in hot vents are very heat-resistant for example.
Bottom line is 'how much would it cost and how much could you make
from stable enzymes?"

Enzymes are polypeptide(s) that are folded in a specific way given the
biochemistry (pH, temperature, ion concentrations, other stabilizing
proteins, etc) of cells. Because they are adapted to such a narrow
environmental range, it is unreasonable to expect them to be capable
of performing their given functions upon being transferred to less
optimal conditions. Ever wonder why fevers make you feel so miserable
even though one could objectively say they only raise your body
temperature by a few measley degrees. Well, those few measley degrees
are a huge deal for the human body whose delicate cellular structures
require extremely narrow optimal environmental conditions when
compared to something like a car which is much hardier in terms of
operating temperatures.

Also remember that enzymes/proteins in general are temporary
molecules. They are not meant to last a long time the way one would
expect a wooden chair to--rather, our body is constantly producing new
proteins while degrading old and damaged ones. So even if our own
cells are incapable of maintaining a particular enzyme for five
months, it would be a lot to ask for us to perform what nature cannot.

This is assuming you want an enzyme to be actively catalyzing its
reaction the whole five months. We are capable of storing protein
solutions for long periods of time by freezing them. A lot of biotech
companies produce various enzyme products for R&D that can be stored
at -20 to -80 Celsius for months and even years. ie, Taq polymerase is
a common enzyme used for research, forensics, etc., and I have used
kits containing it that were viable after sitting in a freezer for
over a year.

As for potential uses... well, that's stuff for another time.