While 3-betting pocket sevens is a strong preflop move, over-barreling on boards that favor the caller's range turns a marginal pair into a low-equity bluff.
Checking is the preferred strategy here to realize our equity and protect our checking range on a board that connects well with the callers' ranges. By betting, we bloat the pot with a marginal second pair that struggles to improve and often gets called by better 9x or draws.
Note: Betting is too thin here; checking allows us to control the pot size and realize equity with our marginal pair.
The Ten is a poor card for our range and improves many of the hands CO calls the flop with, such as JTs or QTs. While we pick up a flush draw and a gutshot, our hand now functions better as a check-call candidate rather than a multi-street barrel.
Note: Continuing to barrel on a card that favors the caller's range is overplaying our hand's strength.
The river completes the flush and pairs the board, making our 7-high flush a strong but cautious value hand. Checking is a viable mixed strategy to induce bluffs or control the pot against potential full houses.
Facing a small bet on the river, we have an easy call with our flush, though the solver prefers an aggressive shove to maximize value against worse flushes or trips. At NL200, calling is a disciplined way to avoid running into the top of Villain's range on a paired board.