Backpacker
I cannot agree with your Interpretation of the safety benefits of a twin.
Yes climb performance especially at grosse weight and high temps can be minimal or nil and yes a lot of accidents occur with pilots attempting to climb on one engine.
I had an engine failure on a 100 hr total time Seneca but still had at my guess 30 % power. I did not shut the engine down until I got up to 1000 feet.
I held one hand on the prop lever ready to pull it in event of a complete failure. At 1000 feet I shut it down flew level and landed.
Cause three broken rocker shafts!
A light twin does badly climbing on one but is brilliant flying level on one.
15 plus years ago we shut an engine down flying Bournemouth to France and didn't restart until approaching the French coast.
Not a clever thing to do but it does show that a light twin is perfectly happy flying level on one.
There is no reason at most fields why you cannot go for level flight at 500 feet rather than a climb. Then once established in single engine cruise step climb on the trim very slowly to circuit height.
At night, over fog in IMC or over water a light twin will fly well on one but in level flight.
Maybe training should look at light twins in a different light.
The light twin gives more options than purely heading for the ground as in a single. Light on fuel and PAX it will climb so yet another option.
Pace