Consider the Motivation of the Ghost
The lore of the ghost, in the Monster Manual, tells us (p.147):
Unfinished Business A ghost yearns to complete some unresolved task from its life. It might seek to avenge its own death, fulfill an oath, or relay a message to a loved one...Others are driven by wickedness or spite, as with a ghost that refuses to rest until every member of a certain family or organization is dead.
The ghost has business to attend to. As you correctly point out, if it possesses a PC and attacks the party, it is going to be pretty ineffective. So why are you having it do that? Ghosts are not designed to be used as challenging combat encounters for the party1.
If, in the unlikely event that the ghost's own agenda is served by attacking the party, it needn't use possession - its withering touch is going to be more effective than a single, non-proficient attack, especially if its host body is in armor1.
Why is it possessing?
If possession is not an effective combat tactic, why is the ghost able to possess? Returning to the lore (emphasis mine):
A ghost is the soul of a once-living creature, bound to haunt a specific location, creature, or object that held significance to it in its life.
Ghosts, in their natural form, have great movement capabilities - a fly speed, incorporeal movement, etherealness. Yet for all of this they have an important restriction - they are bound to a specific person, place, or thing and must remain in proximity to it. They cannot leave the area. Thus the primary reason for them to possess a body is to be able to leave the region they are haunting and go somewhere else to complete their affairs - they would generally not be using the body to attack the party.
Alternately, they may already be where they need to be, but be unable to complete a task without a physical body. If they need to get an object out of a closed box, for example - they can reach through the box, but they cannot open it and take the object out without possessing a body. A ghost should be possessing PCs only when doing so allows them to complete their unfinished business, and it is likely that their lack of proficiencies will not be a hindrance there.
Am I limited to swinging around a spear, once per turn, with disadvantage, until the Paladin is knocked out?
No, you are most certainly not limited to that. You can have the ghost simply dash off to attend to its own affairs, leaving the party to give chase. You can have the ghost patiently explain to the party that it is going to need the body of the paladin for a bit, and ask for their help. You can have the ghost threaten the party with harm to the paladin's body, or with forcing the paladin to break their Oaths, if the party interferes. You can have the ghost pretend to be the paladin until a more opportune time for it to slink off without drawing their attention. There are plenty of options better than immediately using a non-proficient weapon to attack the party for no reason.
But if you must
An encounter with a ghost will typically be a social challenge, as the party attempts to discover what the ghost wants and then decides whether they are going to help it or not. If for some reason you want to make the ghost a combat encounter, then as you correctly cite, the possession feature doesn't give the ghost access to the host's proficiencies. But nothing prevents the ghost from using its own proficiencies, and the equipment section of the MM says that we can:
Assume that a creature is proficient with its armor, weapons, and tools.
While no armor or weapon proficiencies are listed in the ghost's stat block, we can again look at its lore2, where we are told:
A ghost is the soul of a once-living creature...A ghost might not realize that it has died and continue the everyday routine of its life.
Whether it realizes it is dead or not, a ghost remembers its former life. In fact, the reason it is trapped in its ghostly form is that it is uniquely unable to 'forget' its former life, to leave these memories behind and move on to the next stage of its existence. Thus it is entirely reasonable to assume that it retains its own weapon and armor proficiencies from when it was alive. If it is possessing one party member for the express purpose of attacking the others, it should be selecting a party member whose available gear matches its own pre-existing proficiencies, so as to be able to use those proficiencies to the best effect while in its host body and with the gear that the host has.
1Suggested reading: The Monsters Know What They're Doing. Undead Tactics: Ghosts and Mummies.
Some selected quotes:
A ghost will possess a player character in order to perform physical actions that it can’t perform in its incorporeal state (such as opening a door or a container or retrieving an object), to talk to people to whom it doesn’t want to reveal itself as a ghost, to move outside the place it haunts, and so forth...
In short, ghosts aren’t interested in combat unless it involves killing someone they’re compelled to take revenge against. They’ll fight back with Withering Touch to defend themselves, but as soon as it’s no longer necessary—or they’re hit with damage that truly hurts them, or turned—they’re done.
2While it is unusual to need to refer to a monster's lore in order to effectively run them in combat, it is not unheard of. For example, if the players held an undead or golem underwater in the course of a combat, a DM would need to refer to their lore to realize that they do not need air and the suffocation rules do not apply. That information is not contained in their stat block.