2013 500 POP Does not start

Well, after nearly 96K miles with zero issues, the day came that my beloved
Red Thunder, is having some troubles.
When I turn the key in the ignition, i hear the usual buzzer, all lights come on,
and that's just about all.
NO clicking sound repeated. No trying to start, nothing.
I went to Autozone and OReilly and they scan the battery and said it's good,
a little old (only 25% of life left) but as a battery looks still good.
Some times it starts normally, with no issues. When it does not start, i do the
usual jump out, push, jump in get it in 2nd gear, and let go of the clutch (yes,
it's a standard, what else..?) Yes I am that old that I can remember how to do
that, and I grew up in Italy driving the original 500 doing the same things..
When I do that, it starts right away, and then works and hums like a kitten
and it's a joy to drive.
Now, I am looking at:
1) Alternator
2)Starter
3)Battery.

Alternator, i read online and I really can't seem to have the symptoms of a
bad alternator.
Starter.. this is possible.
Bad Battery, maybe. I need to try and jump start it and see what happens.
I also think of the clutch sensor being faulty
Any suggestions ? Are any of those two, alternator and starter something
that one can do DIY ?

Thanks for all the advice.