Steelswarm Gatekeeper only provides an extra summon after tribute summoning the first time.
Bug reproduction steps
Have Steelswarm Gatekeeper and steelswarm monster in play, without using your normal summon for the turn. Tribute summon a steelswarm card from your hand, not using the gatekeeper as a tribute. Repeat this step two more times.
Screenshot OR error code
As long as Steelswarm Gatekeeper remains on the field, and you have only used your normal summons to tribute steelswarm monsters, you should be able to normal summon.