why didn’t the government hand out weapons to civilians?

That's precisely what the "Dad's Armies" were. The Local Defence Volunteers, later the Home Guard, were formed precisely to be a civilian militia to defend against invasion. And they weren't small: by the end of 1940, there were 1,200 LDV Battalions across the country.

Before the formation of the LDV, there was infact a great deal of debate as to whether - and how - a civilian defence force might be formed. Various proposals were tabled, from distributing military weapons to pre-existing rifle clubs, to handing out rifles to golfers, to giving out weapons to anybody who asked for a weapons licence. Indeed, in some areas on the East Coast, ad hoc militia units were already organising and arming themselves.

As you might expect, the government and armed forces had several reservations; primary concerns were that without any training or supervision, overenthusiastic but untrained volunteer units were just as likely to hinder as help effective resistance operations. Concerns were raised, especially during the Battle of Britain, that overzealous volunteers might attack parachuting RAF pilots, especially given the numbers of Eastern European pilots in some RAF squadrons, believing them to be infiltrators, and there were also worries that, if the Germans were to invade, armed civilians were likely to be shot out of hand by German forces if captured rather than treated as POWs, which would also then increase the potential for German attrocities against occupied civilian populations. Many ministers and armed forces officers also expressed serious misgivings about distributing military-grade weaponry into the general populace without any training or oversight, especially with rifles and other small arms in relatively limited supply following the Dunkirk retreat and mass mobilisations. Indeed, many LDV units were originally armed with Canadian Ross rifles, highly accurate but ultimately unreliable weapons that had been withdrawn from frontline service by the Canadian Expeditionary Force in 1916.

The organisation of volunteers into the LDV units allowed for some attempt at the standardisation of training and equipment, the provision of uniform and enough government oversight to make the militia actually useful.