Politicians take the easy way out all the time, and guy buybacks and a reduction in firearms ownership with simplistic rules is what happens every time.
The politicians need to develop a firearms ownership system that contains intense examination of people wanting firearms, their character, what is driving their "need" to have a firearm (or multiple firearms), their social behaviour history, their attitude towards women, and whether they suffer from extreme jealousy and view women as chattels, and any record of beliefs or religious views that contain specified hatreds.
Add in any record of depression involving suicidal thoughts and firearms, and stop all those who present with behavioural problems, from acquiring firearms, and that will solve a lot of the problem. Muslims who believe in jihad and fatwas need to be very closely monitored, and never given a licence for any firearm.
We already have laws forbidding criminals from gaining firearms licences, the final step is doubling the efforts to prevent possession of illegal firearms, and to prevent the illegal firearms trade. Double the sentences for being in possession of illegal firearms. Add possession of illegal firearms to the terrorism laws, treat illegal firearms possession like the terrorism it is.
The links between major violent crimes, the drug trade, and illegal firearms is already well known, take these people out of society for much longer periods.