DV is right to focus on this IMO. There's a certain amount of assumption and inconsistency in either account re the keys.
It's not clear to me that the people who said they saw her take the keys actually saw this, or just assume she must have done because she'd need them. If she did take them, then you do have to wonder how the police got in. The owner was a helicopter pilot in the Middle East, which is four hours ahead. If the only keys had vanished along with SJL, then somebody from Sturgis would have had to phone him, catch him on a reachable number at 11pm local time possibly waking him up, explain that Sturgis had lost his keys, and could he please phone whatever neighbour has the spare set and get them to hand them to the police. Alternatively, nobody bothered, the police just broke his door down and somehow it got repaired. Nobody mentioned any of this at the time nor does anyone now recall any of this having to happen. The detective whom DV and his assistant met conceded he didn't know where they got the keys. Furthermore, if SJL did take the keys and did go to 37SR, why didn't she use them to go inside (fingerprint evidence says she didn't), and why did HR later claim to have heard people leaving?
Arguing that she did take them is people saying she did, and witnesses claiming to have seen her outside the house with them. You are then back to wondering who got in how. DV spoke to the two officers who went there first and they reckon they didn't go in till next day. If she was seen outside 37SR with the keys then you have to dismiss the 123SR sighting as well because WJ's account implies that SJL was never at 37SR.
Another anomaly is that the police broke down the door of her own flat even though they had PL with them and were in contact with her family. Surely her parents had a spare key to her flat? If not, then it tells you the police weren't shy about busting doors down if they had to. So if they didn't have the 37 SR keys, they'd surely have just bust into there like they did into her flat.
I would dismiss the idea that there was one crummy Yale lock on the front door of 37SR. I moved to London in 1988 and you flat-out could not get insurance on your house contents without three bar mortise locks or whatever they're called on every door. If all you had was a Yale, no insurer would even quote.