Radio does not work.

the dash lamp IMMO will not lie, it tells you if the key is good or now, covered in the free users guide.
but use the IMMO can lie of battery is near dead. sure.
not told what the gear shifter does,
what does CUT ON mean? (radio dead or works/)
the steps are easy cart like this auto transmission
we can crank the engine in park or neutral if we step on the the brake pedal, the lever moves. at your demand, called park lockout feature
I get it ,does nothing means the starter is 100% dead
while key turned (cranking) in PARK.
turn the key to on /run position the IMMO lamp talks to you now, it tells 2 things key good or key bad
if bad no immo car will ever start. or starts for 1 to 10 seconds and stalls.
if the battery is dead or failing many things will word and not work, for example radio works but the starter is dead the starter uses 1200 times more power than the starter.. so radios can work. with near dead batteries.
for 70 years true.
so, check battery lugs for rusty or loose
charge the battery and load test it.(in a shop)
or buy the tools.
or jumper cable jump the battery.
or call AAA auto club tow,and say jump start me car.

also learn that IMMO can lie to you if battery is weak.
answering for NO CRANK cars.

That battery charger is not any kind of battery tester, it can not say by its self good or bad battery. ever.
only load testers can or the below. (cranking tests)

it my get correct battery is low and needs to be charged
the charge can be used (good ones) force to charge the battery a 15amp charger is best, (2 hour can cure this)
after the battery is force charged or jump cable jump charged, the stater motor nmnow starts the engine
this act takes, 100 to 300 amps of current flow, and proves batter is both good and bad, or load test the battery in any
autostore.
do not leave cell phones connected when car is not running or soon.
the charge ports on my car , one is hot full time key off, so never leave things on any hot all the time power port.
next up is BCM games (facts of life) and worse now with 10 computer in cars.

The BCM brain the body brain , will go crazy when the battery does drain low (or battery bad) at near 10vdc
some electronics go dead or random. acts of crazy
all computers do that, as there, regulators chips go offline or bounce at the threshold of power loss,(weAk)
all computers (processors and micro controllers do that ) bad power DC.

test 1: is charge battery first fully, and start engine
battery voltage (a dmm meter) reads 13v to 15vdc, lower volts is bad alternator (or its 3 fuses blown) one of them.

test 2, for phantom drains test short version ask for 10- page versions.

KEY off, battery fully charged use a real battery charger, 15amps is good and forced to charge.
ammeter clamped to the battery PLUS lug wire large (a tool)
current flow key off spec max is 0.050amps (50mA) or less. over is called a phantom drain, (or current)
ok it fails
in the dark (totally) find lamps stuck on in the car ,every where. even under hood lamps ,glove box. etc.
the silly cell phone, uses 1 amp to charged 1000mA, and is 20 times over spec , key off so remove he silly phone.
cured? (a shocking discovery , LOL)
NO !
same ammeter showing the current way too high key off
NEXT:
pull fuses , 1 by1 , one out, see no drop, fuse back , next fuse. find the fuse that causes huge drop in current.

this finds the branch of the wiring tree that is overloaded.
most times this finds the path that is draining. (some alternators do this and fuse to it can not be found easy)

to go to the end run, ask, ( all electric tech know this process)
ask.

2006 Trailblazer won't start - Discussion on Topix

www.topix.com > Chevrolet TrailBlazer

Sep 6, 2012 - My 2006 Trailblazer will not start, bought a new battery, still will notstart, not even with a jump. ... i get a jump i drive it and charged the battery it seemed fine but then i went ... Then, next morning, when I turn key, all lights come on and radio... back a day and 360$ well that pos still will crank but will not start. 02 trailblazer a/c problem
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Check alternator voltage output, might be low. obd2 scanner

looks like a dead battery to me

Not the Key but the ignition itself. Replace teh ignition module where the key is inserted to start the vehicle and that should resolve the issue.

check the fuse under the kick panel

Yes. Because the security system was "set". You need the code to unlock the security system for the radio. It is like that for theft pruposes. Best bet is to go to the dealership to have them reset it or figure out how to get the code for your radio.

both,the battery has been run down too many times and your radio have a heavy power draw,if you have an aftermarket alarm it may have something to do with the problem.

battery is dead... have the battery and alternator tested for functionality