Does removing a unit in reaction chain prevent its when I'm played ability from activating?

No, removing a unit in response to its "When I'm played" ability does not prevent that ability from activating or resolving.

According to RiftJudge FAQ #8020 and FAQ #5217:

• "When you play me" abilities (Play Effects) still resolve even if the unit is killed, bounced, or banished by a reaction to that ability.
• Once a unit with a Play Effect is played, the unit enters the board immediately. The Play Effect then triggers, enters the chain, and provides a window for players to play reactions.
• Triggered abilities resolve independently of their source. While effects within the ability that reference the unit itself (e.g., "this unit," "here," or location-based references) may fail if the unit is no longer in play or at the correct location, the rest of the ability's instructions will still execute to the fullest extent possible.

The Sequence

Following RiftJudge FAQ #6781 and Core Rule 383.4.a.2, the sequence is as follows:

1. Play: You play the unit card. It enters the chain and is immediately finalized to the board.
2. Trigger: Because the unit has a "When I'm played" ability, that ability triggers and is placed on the chain as a pending item.
3. Reaction Window: The state is now CLOSED (a chain exists). Players can now play [REACTION] cards to respond to the triggered ability.
4. Removal: If you use a reaction to remove the unit, the triggered ability remains on the chain.
5. Resolution: When players pass, the chain resolves (LIFO). The triggered ability resolves, performing as many of its instructions as possible.