There are indeed many things about the US system that treat foreigners as shit with respect to visa access. But none of that is about giving fewer constitutional rights to foreigners who are physically inside the US.
Visas only control the permission to travel to the US and to apply for admission at a port of entry, and all visa applications and decisions happen while the people taking the action are outside the US. (Visas are only formally applied for at the visa interview, which is never conducted in the US, and visa denials are also issued by the relevant US embassy or consulate abroad and not by the Washington-based part of the State Department.)
The continued legality of anyone’s status within the US following a legal entry does not depend on their visa remaining valid or unexpired. Even if their visa is revoked while they are in the US, that itself only affects future entries, although other actions might of course be taken by the government in parallel.
Importantly, the right to be allowed to enter or remain in the US is one of the constitutional rights that only exists for citizens, no different from most countries. I did say that those few rights which are inherent to citizenship don’t apply to foreigners inside the US. Again, I know of no country in the world which gives foreigners a right to enter and remain, except for conditional rights for specific nationalities like with EU freedom of movement.
Here are two examples of how foreigners have more (and approximately full) rights while inside the US:
Foreigners in the US have far more due process rights if the federal government starts a factually or legally unjustified removal proceeding against them than if a visa officer abroad makes factual or legal errors when denying a visa. The former scenario can often be challenged in administrative, quasi-judicial, or actual judicial proceedings; the latter case has no remedy except to persuade the State Department to change its mind.
If the US government sniffs the communications of two foreigners located within the US, they need a warrant under the same conditions as they would if the foreigners were citizens, unlike the case of purely external communications where the US government is only so constrained if they know a citizen is involved.
And so on.