TLDR: Conscripts lose in long and drawn out battles of attrition.
Well the conscript is not unlike the King Tiger if you think about it.
On one end of the unit spectrum you have the King Tiger with it's highest price and highest strength of all units in the game.
On the exact opposite end, you have the conscript with it's lowest price and lowest strength of all units in the game.
Like the King Tiger, dealing with a unit on the extreme end of the spectrum requires a certain strategy.
Quite often, the strength of conscripts is in their number, there's not just 'one' conscript unit. You can quickly turn this into your advantage with surrender mechanics.
I play on a special difficulty setting, which means the normal 15 strength enemy conscripts are 20 (twenty) strength units.
One shot from a powerful piece of rocket artillery, or two from lesser artillery types will give you this:
As you said yourself:
the most you can hope to kill or suppress is 10 points
Which is exactly what I accomplished, 4 killed and 7 suppressed for 11 total points.
A follow up from a Panzer IV will give you this.

And just like that, a 20 strength unit is finished off, without ever managing to have shot back at all.
Sure the conscript is much cheaper and easier to replace, but clearly this is a futile effort. Each conscript has dealt no damage in return for it's destruction, so it is only a matter of time before the more expensive tank and artillery combination pays for itself in terms of conscript units destroyed.
The same holds true for multiplayer, not just against the AI.
If all you do is buy the most expensive tank, the conscript is a serious problem. The better strategy is to make more use of the medium tanks (which are being vastly improved in the upcoming patch as well!) to handle soft targets.
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In this game, my opponent tried to conscript flood me. As you can see from the final kill tally, I destroyed a total of 77 infantry units.
So for a very long time, yes I had to constantly kill wave after wave of conscripts. But while I was doing so, because I was using abundant artillery and medium tanks, I was never losing any prestige in the process because of the minimal damage my units received. The proof is right there, 1 total tank unit lost, 1 total artillery unit lost compared to 77 infantry units(seventy-seven!!).
At the same time, each conscript I destroyed needed to be replaced and repurchased, which slowly but surely drained my opponent of all of his prestige.
After all, the Soft Attack of a King Tiger is only 9.
The Soft Attack of a lowly Panzer IVD is 8.
The price difference between the two is absolutely enormous!