During your Main Phase: You can target 1 "Scrap" monster in your GY: Special Summon it, then destroy 1 card on your field. If a "Scrap" monster(s) you control is destroyed by a card effect: Special Summon 1 "Scrap" monster from your Deck, then destroy 1 card on the field. You can only use each effect of "Scrap Wyvern" once per turn.

The text uses then which means the effects to special summon monsters and destroy a card are not simultaneous. The last thing to happen is the destruction so this should be fixed.

Ex soul take causes things to miss timing because the last thing to happen is lp gain.

Target 1 face-up monster your opponent controls; destroy that target, then your opponent gains 1000 Life Points.
Works fine for me, used Scrap Recycle and it could not activate its effect when it came back as it missed the timing. Are there any cards in particular that did not work as expected?

Please provide a replay showing the effect not working as it should, also please state on which server, game version, and OS this occurred on.
I was testing using the FF xyz, not sure if it behaves differently with materials but I see no reason why it shouldn't trigger again. I also have many targets in deck. 

pc, windows 7, LAN
This is nothing to do with Scrap Wyvern and everything to do with this:

Brotherhood of the Fire Fist - Eagle Was on the field allowing you to use Brotherhood of the Fire Fist - Tiger King's effect even though you could not send enough Fire Fist Spell/Trap cards to the Grave.
By the time you use Scrap Wyvern to repeat this, you no longer had Brotherhood of the Fire Fist - Eagle on the field, thus had to pay the cost to use the effect. But you still did not have 3 Fire Fist Spell/Trap cards on the field to pay. So you could not use the effect.

Side note:
Please attach your replays directly to your posts using the tick box just above the post button as you are making your post.
I didn't have three in that replay but the issue was I never had 3 face-up. I didn't think you'd need them face-up to trigger him but ocg ruling says face-up for some reason.
You never had 3 Fire Fist Spell/Trap cards on the field. The only reason he worked the first time was due to Brotherhood of the Fire Fist - Eagle's effect. As he was not there the second time this is your issue.