What “good old fashioned police work” really means is due diligence. It’s the same reason why you shouldn’t just extract mobile data in a homicide case without also obtaining call detail records from the service provider.
So rather than try to rationalize the lack of budget at a press conference by talking about how the agency can get the job done without technology, consider talking about the number of hours that went into an investigation, why this is suboptimal efficiency, and what technology would help. Take this message to the city council or the county board; show the numbers, compare with numbers from neighboring communities that are using technology to their benefit. Convince them that the purchases will attract the people skilled enough to use them--and that this will ultimately benefit your community’s need for the best policing their tax dollars can get.