They won't hit a "random target", they'll hit things like airfields, command centers, and ammunition dumps.

For example, a strike could take out several of the jets the terrorists in the Kremlin are using to kill civilians in Kharkiv.

Long range attacks are about knocking out supplies, the ability to produce new supplies, and the ability to get supplies to the front line.

Say Russia is getting more newly made artillery shells to the front line. The best way to fix this is to blow up the factories.

If you can't take out the factories directly, take out components that the factories need to operate: ore processing, fuel refineries, electricity grid, etc.

The problem is all the critical targets are a long ways from the front line and Ukraine currently has limited capacity to hit them.

Meanwhile Russia is targeting all of Ukraine's internal infrastructure constantly.

Long wars are won by the production capabilities of the groups involved as much as the front line troops.

If they were random targets, sure. But the messaging clearly included the wish to strike military targets at long range. Don’t have to lose people to a aerial bomb if the plane carrying it has been destroyed by a long range strike.

It slowed the progress by necessitating the movement of invading troops back to defend. Ukraine also had a major propaganda win from the Kursk offensive.

What benefit does striking some random target hundreds of miles within Russia accomplish

They’re not random but attempts to make more strategic difference, and to expand the war beyond just the front

• how can Russian artillery keep shelling if their supplies are blown up, and the supplies for those? And how responsive can they be at re-supply if new supplies have to come from hundreds of miles?
• how can Russia keep feeding the meat grinder if fresh troops need to fight their way to the front, lose their supplies, and take losses even before they get there?
• how can Russian commanders work if they’re dead? And their commanders are dead? And someone is trying to make battle decisions from hundreds of miles away?

Think of the Russian Black Sea fleet. The surviving ships are so far away that they’re not making any contribution to the war. Now, imagine making the Russian Air Force ineffective, Russian Command ineffective, and the supply situation ever worse

What benefit does striking some random target hundreds of miles within Russia accomplish for Ukraine, who is slowly losing the ability to hold ground?

The goal of the Ukraine conflict is to create a Pyrrhic victory for Russia so costly that it collapses from within. Long range missiles capable of destroying key domestic infrastructure and material reserves will (hypothetically) hasten that economic collapse.

If you can start hitting major financial centers, administrative offices, and social meeting spaces, you might even be able to decapitate some of the senior leadership and kick off a succession crisis in Putin's government. Alternatively, you push these leaders so far down into hiding that they can no longer effectively govern the country and foreign-sponsored insurgents can take over.

That's the theory of the war, at least. In practice, NATO leadership is still unsettled by the prospect of a Total War with a nuclear armed state. So they're not quite ready to drop a long range cruise missile on the Frunzenskaya Embankment, for fear of what a retaliatory strike would look like.

Everyone talked about tanks as ww1 winners. But people don't quite realise the stupendous artillery advantage the allies had. 2 or 3 to 1 by the end of the war in places. And significantly more shells. There's a reason we are still digging then up today

That doesn't make any logical sense. You say it's a financial strain to send equipment and propose sending NATO in to help. Where is NATO getting their equipment from?

ITS THE SAME EQUIPMENT.

Sending NATO in fundamentally breaks the treaties upon which the organization was founded. NATO isn't a joint military force to send to places for freedom and democracy, it's a defensive organization. Without an explicit article 5 declaration from a current member state against a specific threat for a specific reason, a deployment of NATO would be undermining of the organization's mission in itself, putting in question whether NATO is there to protect its members or just to be used as a cudgel for potential inconvenient threats because the alternative is a financial strain.

Interventions like this, such as the 2011 intervention on Libya to enforce UN resolution 1973 will fuel arguments that NATO is altering its mission from defending its members to participate in US led (or in this case also French led) interventionism. Upon such a time where threats are no longer direct as is the case of the current Russian invasion of Ukraine, this will have implications on the future of the organization and questions will be made about its purpose, as it has happened before after the fall of the USSR. Ultimately NATO's mandate must be upheld and strictly followed, for the sake of the treaty. At least that's the opinion i share.

Agreed its time to send NATO in. NATO should have closed Ukraine's airspace when this started.