No.Cranking slow is exactly the reason to look at a hot start kit.
Thats why they were invented
If the starter motor RUNS AT ALL, the solenoid is engaged, and that means there's enough power to the "S" terminal.
When the starter DOES NOT CRANK, the "hot start kit" you describe may be useful.
Perhaps I missed it:
What is the battery voltage when the engine cranks slowly? For that matter, what is the battery voltage with no load? A fully-charged "12-volt" battery had better have 12.6--12.7 volts with no load, and at least 9.5 when cranking, but more likely ~11 volts. More is better.
Frank Einstein mentioned voltage-drop testing; but neglected the ground side. Here's a diagram:
https://www.chevelles.com/media/starter-voltage-drop-tests.48355/full
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