A very sad but interesting video.
OK, here is how I see it.
Aircraft suffers a single engine failure at a weight that is too heavy for continued flight - committed. Ac does exactly what it says on the can and moves rapidly towards surface. Pilot attempts to cushion impact [conning angle] and makes a reasonable go at the landing - from my own experience it happens at exactly the rate seen in the video [although mine was on to land!]
Ac next seen pitching fore and aft. He probably didnt dissengage the stab system whilst on the water. It thus tries to fight the swell and it got out of phase. It probably stopped when he dissengaged it.
Pilot then attempts water take off. Advice is to get ac up to speed on the water before drooping NR to lowest in flight value to give the required energy to break the water suction. Way to judge this - bow wave just going over the windscreen. Look at the video, that is when he rotates forward. Bet there is a note in his manual that says if the nose bay fills with water it may not be possible to complete the take off! What you do not see is if he tried to reduce the weight for take off. If it wont hover on both it will not get airborne on one! The other thought was, why was it being filmed? There is an awful lot of safety style boats around etc. Was this a trial of its water take off ability?
Undercarriage. 2 thoughts. One, on impact the gear selector is forced down by the impact - without the pilot realising?[lets face it, he is a bit busy about then!] Second, he puts gear down for stability during the pitching about in the swell. I suspect he probably meant to retract it before he began the water take off.
All in all, very sad. Thoughts with family.