Usually, we don't use "since" in the simple past. However, there are a few specific cases in which we can use it as follows:
1- We can use it as a time adverb such as it's long/a long time since I met him,
it's two weeks since I joined the bank, etc.
2- We can also use it as a conjunction where it means "because" such as since you
don't have money, I'll not take you to the market, you should take Mr Adam's
permission since he is our General Manager.
As for the sentence "Deputy chief constable fights against crime since he was first appointed", I don't think this is acceptable although some people contend that when you have been doing an activity regularly for a long time, you can use since in the present simple for this activity. I don't think it's valid justification.As a matter of fact, the right sentence is "Deputy chief constable has been fighting against crime since he was first appointed".