No, the capitalists can't consume the surplus product. This has been considered and it has been proved not to work (eg by Luxemburg). The reason is, that capitalists are in competition with each other and are thus forced to invest in constant capital (machines etc) expanding circulation and making the problem worse. Capitalists who only consume the surplus are outcompeted and cease to be capitalists.

One real way that this contradiction is (temporarily) dealt with, is to expand to markets who are not yet fully under the capitalist mode of production (colonialism/imperialism). When England ran into this problem with their textile industry, they destroyed Indias textile industry, forced them to buy English textiles, forced them to sell opium to China to pay for the textiles and forced China to buy the opium and pay with silver that the English wanted. There aren't many white spots left on the globe though so this "solution" is starting to run into problems.

Another tried and proven way to deal with the crisis of overproduction and get rid of the surplus product, is to destroy lots of value. Like in war. This unclogs the arteries of capital circulation and gets things following again. The state can help expropriate people via taxes or inflation and funnel that directly to the military industrial complex. Crucially, this has to happen without providing any use value to anyone or it would cut into profits. So the state funding weapons to blow stuff up works, but building useful things like schools or railways or housing doesn't, because it would lower the profits of private companies.

Another big one is debt and financialization. Just gamble on future generations being able to afford the present surplus product. Debt is a claim on future labor, so to realize it, the productivity of labor has to go up. But oops, this again increases the surplus product. So no permanent solution.

The only thing, that genuinely works is war. The more destructive the better. Or, you know, getting rid of capitalism.