So the Xbox One launch had a ton of issues that have been well discussed. These range from the high price, the forced inclusion of Kinect, the lack of enticing launch exclusives, the staggered worldwide launch due to Kinect, etc.

One aspect that I find really baffling is the lack of backwards compatibility with the Xbox 360 (and OG Xbox). The PS4 did not have backwards compatibility but that was somewhat understandable because Sony smartly moved away from the Cell architecture towards something more multiplat friendly. I am less clear on why Xbox One did not have BC on Day 1. This is even more surprising because Xbox One eventually gained limited BC down the line through software emulation. So it was possible. Was Microsoft wrong for not prioritizing backwards compatibility on day one?

Think about it, the 360 had a huge user base (especially in USA) and a great library. In the early days of that gen when the software lineup was lackluster having access to the 360s best titles could have made Xbox One more appealing. MS execs have since gone on record to state that the Xbox One generation was the worst one to lose because that's when digital game sales took off and players became more locked in with separate ecosystems. 

Then again, maybe it would not have made a difference. After all, the Wii U was fully compatible with the previous gen leader the Wii and that did not help its sales noticeably. 

What do you think?

I don't think it would have made much of a difference to be honest, the damage was already done as far as the system being weaker and more expensive than PS4, plus the radioactive PR surrounding it.

A PS3 style turnaround was maybe possible if they'd managed to deliver killer games, but they failed there too.

Backwards compatibility is awesome, but it would have barely changed the first 2 or so years of Xbox One. Firstly, this isn't a PS4/PS5 situation where over 99% of titles are supported. But even if I entertain a hypothetical layer to this discussion that Xbox One plays 90-99% of 360 games, I still don't think it's enough of a selling point.
Xbox One had the best Xbox launch in history up to that time despite the negativity. In all fairness, Xbox reversed many of the worst aspects of Xbox One but there were still issues.
Xbox One might sell a million more a year if backwards compatibility is a thing from the get-go. It might finish with 61-65 million units in the long run.

PS3 had day 1 BC with PS2 and PS1, didn't help against the 360 in the early days.

BC is great, but not a selling point for a new console. Why spend a ton of money just to play the games you already played.

It does help when switching over or buying a secondary console as now you have access to all you 'missed' in the previous generation. But that's fine halfway during the generation. Early adopters are usually the loyal fans looking for 'next generation' stuff.

As you implied, lack of BC on day 1 was the least of One's problems.