IS tthis a DIY question? or just want guessing? ill pretend DIY/.
no, the car will not crank.
cranking uses 100 amps of current,
the HL , maybe 10amps, 10x less.
turn over means nothing, cranking matters
both the starter and engine spin, "turns over": (1 can and other might not)
starting is a whole other matter, as is sustained start, called running.
crank?> start, started, running, 1,2,3,4.
the heater blower runs, key on, but the heater switch is off.
seem like the controller is tad upset, (lets fix engine first) and fix blower later. (my bet is it wakes up when battery is happy)
my guess is car has and automatic transmission, right>?
and as such to crank it must do so in, park and in Neutral.
does the engine crank in NEUTRAL. (hold foot brake, keyon, shift th N. ) crank now?
1: check fuses yet?
2: check rested battery voltage yet (meter) 12.6vdc.?
if less battery is bad, or only discharged. if yes, charge it?
no charger use jumper cables to crank car. got cables?
call AAA tow they jump start it? no AAA?
barrow a charged battery and try that, ok battery is charged and it will not crank,,, ,good progress now.
3: i next hot wire the starter solenoid. (some cars this is very hard)
check for about 11vdc at starter key line cranking.
0v means its dead.
hot wire that small starter wire, there, cranks now , key off
the starter is good.
now check voltage at the left side of P/N switch, Park/neutral.
12vdc id good, cranking, 11 is too.
if you cant get at least 11v out of switch 20. that means
the P/N switch is bad, (11 or more input)
(battery is good, solved that in test 2)
so bad transmission park neutral switch called the KEY line.