When I pass priority in a showdown that already declared attackers and defenders, Rumble Hotheaded is removed by a Hidden Blade. The question is, is the assault buff given by Rumble removed with it, or are my other mechs still buffed?

When Rumble, Hotheaded is removed from the board by Hidden Blade, your other Mechs immediately lose the Assault keyword.

Reasoning

• Continuous Effect: Rumble, Hotheaded’s passive ability generates a continuous effect that grants Assault to your Mechs.
• Effect Duration: Continuous effects generated by a unit's passive ability are active only as long as that unit remains on the board. Once Rumble, Hotheaded is removed and sent to the Trash, the source of the effect is gone, and the effect immediately ceases to apply.
• Keyword Removal: Because the Assault keyword is granted by the passive ability rather than being a permanent characteristic, the keyword is removed from your other Mechs as soon as the passive ability stops functioning.

Sequence of Events

1. Rumble, Hotheaded is on the board; your Mechs have Assault.
2. Hidden Blade resolves, removing Rumble, Hotheaded from the board to the Trash.
3. The continuous effect from Rumble's passive ability ceases to function.
4. Your other Mechs immediately lose the Assault keyword.
5. If this occurs during a Showdown, your Mechs will no longer have the Assault bonus for the remainder of the combat.

Note: I could not fully verify this ruling against the retrieved rules, as the provided context did not contain the specific card data for Rumble, Hotheaded.