Always check, it might be mate.
The logic is irrefutable since you cannot mate without giving a check. But the rule has much deeper implications. Firstly, it's reminding you that checks are the most forcing moves and should always be considered for both sides. Secondly, it's asking you to check your calculations as a check may not be the best move, giving rise to a follow up maxim
Patzer sees a check, gives a check.
A pawn island majority, that can threaten to promote a pawn, is worth more than the static material count.
https://lichess.org/training/promotion
"The strength of the pawns lies in their ability to advance." (Wilhelm Steinitz)
https://lichess.org/training/advancedPawn
“Give me a queenside majority and the bishop pair, and I’ll beat anyone in the endgame.” (Dvoretsky, Suba, Watson)
https://lichess.org/training/pawnEndgame
https://en.wikipedia.org/wiki/Pawn_structure
Toscani - on a related note, an advanced player at my chess club told me once that a pawn on the 7th rank can be considered the same value as a rook, 5 points
OP - a commonly repeated chess principle is to fight for the center. Protect your king.
Unbreakable principle: King safety and concrete tactics override every other chess rule; if your king is in danger and you ignore the forcing moves (checks, captures, threats), no amount of “good position” will save you.
Give check with each move until mate to not draw drawish positions.