This. Your DH needs to think about his commute and wider travelling requirements.
Does he know where these 'other places in Yorkshire' are that he'll need to visit and at what time of day?
It's just that north Leeds in particular is very congested during rush hour as the motorways for all directions including going north are on the south of the city so there's lots of people driving into the city and also towards the motorways so while places like Roundhay, Horsforth etc are great places to live, it's going to be an utterly miserable commute for him if he regularly needs to be in Sheffield at 9 am.
But then again if he's going to places like Harrogate or Ilkley or only making short trips between 10 and 3 it's less of an issue.
If you have a healthy budget you could look at the villages on the east side of Leeds, Thorner, Shadwell, Scarcroft, Barwick etc, or somewhere like Wetherby which is a nice market town. Also with nearly adult DC you might want to be a bit closer to the city, Garforth will be a good choice. Good train service to the city, although on the east (or south) side the Park and Ride is another good option for commuting.
The weather is significantly dryer in the east than the west, (I think 10 miles east of the city gets half the rain than it does 10 miles west) but I think you're catastrophising about this and also crime levels.
We do get plenty of good weather it's probably not a lot worse on average than the south west which can be quite stormy. Crime is probably no worse than any large city and the worst is concentrated in the inner city suburbs.
But if you've been looking at social media groups specifically for discussing crimes and social problems then of course that's what people are going to talk about.
Leeds is a great place to live, probably one of the best in the UK. There's big city amenities along with amazing countryside. You should consider yourself very fortunate not dreading the move.