The Y chromosome is indeed a small genetic body that has relatively few genes, a lot of non-coding DNA and one important gene called 'TDF' - testosterone determining factor.
Y does not make testosterone - women can make the hormonse too, don't forget (some more than others!). TDF controls when large doses of testosterone are made, and where it is made. Two doses - one early in development and one at puberty - give males their male characteristics and enable their reproductive organs to be able to produce sperm.
With this gene, males are essentially female, as Turners Syndrome (I think it's called this) which is XO, essentially shows.
So while the Y chromosome accumulates silent mutations, TDF genes are not essential for living and can be passed on. If this scientist is correct, we should start seeing more cases of 'females' with XY chromosomes (the Y being essentially defunct as far as TDF goes). There are cases of this, but it is extremely rare.
What will happen if this is true? By that stage, science should be able to solve whatever downfalls. It's a long way off yet - 100,000 years might be a blink in the eye of evolution, but it is an eternity in terms of human endeavour.
In other organisms, the rate of mutation on the 'male' chromosome varies. AFAIK, most other chromosome-dependent species have essential genes on their 'Y' analogues, meaning the chromosome won't disappear.
The question is, how essential is being male to being alive.
Athon