I guess if you're in a Radar environment, you could tell them that
you just lost the GS and would like to get vectors for another
approach. What if there is no Radar?

You have several options.

First off, you can climb immediately - it's only an early turn that is
verboten. If there is no turn, there is no issue.

Second, the GPS will show you when you are at the MAP. I suppose you
could have an ILS with no RADAR coverage and the GPS could fail and the
GS could fail. I think the odds of that are something like "Not in
this lifetime" but let's say it happens.

Even then, it's really not the end of the world. At some point, you're
going to pass over the antenna. When that happens, you are going to
peg the needle. Most pilots can keep from pegging the LOC needle down
to DH, and virtually all can keep from pegging it to about 500 ft. So
somewhere between a mile short of the middle marker and being over the
airport you're going to peg the needle - and then you can start
whatever turn the missed approach calls for, if it actually calls for
one. Unless your airspeed control and your estimate of winds is
spectacular, this is going to be as accurate as timing the approach
anyway.

There are enough important things to do at the FAF - power back, carb
heat, switch to tower/advisory (if not already there) - that I hate
adding one more thing to do for the novice instrument pilot. It makes
it just that more likely that he will forget or screw up something.
Nonessential activities at and inside the marker are bad news.

I am not a fan of increasing pilot workload on every approach just in
case something very unlikely happens. Most IFR accidents are the
result of pilot failure, not equipment failure. I also consider the IR
an introductory ticket. I think most pilots who fly IFR in IMC
regularly will develop beyond the minimum standards of the IR
relatively quickly. When they do, it makes sense to add to their
procedures, which will be no big deal because they will have the cycles
to handle additional tasks.

Pilots who fly IFR only infrequently have so little exposure to the low
probability equipment failures, and so much exposure to pilot overload,
that I really don't want to add anything not essential to their
workloads.

Michael