Are there any index funds that track the entire market? Wouldn’t investing in all companies solve the problem of index tracking?

Yes, there are funds that try to match the entire market. Vanguard runs a couple "total market" indices (VT and VTI, for example), and there are more.

But that's not to say that they actually invest in the entire market. They don't; they use sampling to try to match the performance of the thing they index. (Vanguard, at least, gets very very close to this...)

Disclosure: I am invested in in VTI.