I asked AI and got:
The difference in blood rules between rugby and football (soccer) primarily comes down to balancing player safety with maintaining the flow of a continuously played, highly physical sport like rugby.
The perception that rugby players "play on with open wounds" isn't entirely accurate for modern professional rugby. Both sports have rules to manage blood, but the mechanics of the game necessitate a different approach.

Rugby Union's "Blood Bin" Rule
Rugby uses the "Blood Bin" rule (Law 3.25 in World Rugby regulations) to address visible bleeding:
Temporary Removal: A player with a visible, bleeding wound must immediately leave the field. This is mandatory for the player's safety and to prevent the spread of bloodborne pathogens to others.
Temporary Substitution: Crucially, the player can be temporarily replaced by a substitute. This allows the game to continue without the team being penalized by playing short-handed.
Time Limit: The injured player has 15 minutes (actual time) to have the wound cleaned, the bleeding controlled, and the injury covered with a secure, waterproof dressing.
Return to Play: If the player is cleared by medical staff within 15 minutes, they can return to the field, and the temporary substitute must leave. If they cannot return within the limit, the substitution becomes permanent.
This rule is designed for a game that involves constant, full-contact grappling (scrums, rucks, tackles), which makes small cuts and head wounds common. The temporary substitution minimizes disruption to the game's momentum while still enforcing strict safety and hygiene standards.

Football's Stricter Stoppage Rule
In football, a player with blood on their kit or a bleeding wound must also leave the field. The key difference is the substitution policy:
No Temporary Replacement: When a player leaves the field for a bleeding injury, the team generally cannot bring on a temporary replacement. They must play short-handed (with 10 players) until the injured player returns or a permanent substitution is used.
Return Condition: The player cannot return until the bleeding has stopped, the wound is covered, and any blood-stained clothing is replaced. A simple spot of blood on a shirt is often enough to require a change because of the rule's strict interpretation against blood-contaminated gear.
The stricter, non-temporary substitution rule in football is feasible because it is a lower-contact sport where bleeding injuries are less frequent, and play is already regularly interrupted (out of bounds, fouls, etc.).

The Core Difference: Game Flow and Contact
The divergent rules are a practical response to the nature of each sport:
Level of Contact: Rugby is a high-contact sport where minor head wounds and cuts are highly probable. A rule that forced a team to play short-handed every time a player received a small gash would constantly compromise the balance of the game.
Continuity: Rugby values continuous play, as it has far fewer natural stoppages than football. The "Blood Bin" ensures player welfare is met while preserving the flow and tactical integrity of the match.
Risk Mitigation: Both sports prioritize stopping active bleeding and removing contaminated clothing to prevent the transmission of bloodborne viruses. The difference is in how they manage the player's absence from the field to achieve this.