Renown, because of her speed, was indeed was one of the "best" rebuilds, able to deploy a powerful main armament and useful heavy AA to protect RN carriers. She was still a tinclad however, and even better would have been a rebuilt Hood.
According to the Fleet commander's war diary, quoted in Koop & Schmolke, the first radar detection of Renown was at 04:49 hrs at 20.2 km bearing 295T. Having visually sighted in poor visibility what was thought to be an oil tanker, this was later revised to "HMS Rodney (nicknamed by RN sailors Rodol because she did look a bit like an oil tanker at a distance and RN tankers' names finished in -ol.) This incorrect identification immediately caused Lutjens to commence "running away" at 05:07 hrs when he turned to starboard to 350T having decided his armour was not strong enough to fight a battleship. Renown's first hit was at 05:25 or 05:17 ie long after Lutjens' started evading.
Only after 16 broadsides at less than 16,000 yards. I would expect an old USN BB to shoot at least that good.
Forgetting to mention how bad the weather was- gale force wind and sea with Renown punching into it. I don't believe any
US battleship has ever even engaged
an enemy under such conditions, let alone scored a hit.
Lutjens admits he only turned back 20 degrees to port at 05:14 because his inadequately forward-buoyant ships were "bows under" heading directly into the weather and their A turrets were flooded and useless, he also wanted to steer NE, because Renown could only get her forward turrets to bear- that is the same as running away. During a lull in the shooting at 06:07 he asked Scharnhorst how many enemies there were. "So far only one, she replied" at 06:38. Lutjens wished to report that he was engaged by both Rodney and Renown, but his aerials had been shot away.
With both ships' A turrets U/S, Gneisenau's foretop gunnery control gone and Scharnhorst's engineering plant hopelessly unreliable, Lutjens headed off on a long tour via Jan Mayen Island and Iceland and abandoned Commodore Bonte and his destroyers, bottled up in Narvik to face the music alone.
Lest anyone should think that the old rebuild had outperformed two new ships by too great a degree, one should realise a large forward section of Renown's port torpedo blister had come adrift and the hydrodynamic drag was slowing the old warrior down. This was repaired in Rosyth drydock.
The rebuilt US battleships had effectively been downgraded to monitors for shore bombardment, a bit like the expendable Pre-Dreadnoughts deployed for the same job at Gallipoli. They undoubtedly put up a great show at Surigao Strait, but how would they have fared in a daylight fight against Kurita's Main Force off Samar? What a 1940 West Virginia, not hiding in the darkness, and not equipped with centimetric radar, would have done against the elusive sisters off Stromvaer, who knows?
Footnote: Mississippi and Idaho expended 514 main battery shells against presumed Japanese ships on the 27th July 1943 in the so-called Battle of the Pips, but there was nothing actually there. I am sure Dave can provide more detail
Finally, astonishingly, I really must agree with Alberto. (It had to happen eventually....
) Britain and Italy rebuilt ships because pacifist sentiment, respect for treaties, and lack of budget precluded new construction.
All the best