Why do you think the VAT cut increased economic activity?
Because that's what happened!
Ultimately of course it's impossible to prove causation (I can't see how you can be so sure it didn't), as the control experiment of not decreasing the rate cannot be done at the same time. The purpose of the rate reduction was to stimulate economic activity in labour intensive industries to increase the number of jobs. That was the actual outcome. Would have happened anyway? Maybe, maybe not. There's an interesting comparator, though, at [broken link removed] comparing retail with accommodation and food services.
Would lowering the cost of construction - by removing development levies and/or reducing VAT on input costs - stimulate the industry and lead to more houses being built? My guess - based on the evidence of reducing the VAT rate in another sector - would be that it would. I'd be interested to hear why you are so convinced the opposite is the case.