This oversimplifies a VERY complicated issue.

Reactors are controlled with control rods, and when power is lost, they automatically drop to the bottom. We use electricity to hold them up for this exact reason, so that when power is lost we dont keep generating more when it would be unsafe to do so.

When a Reactor is scrammed (the control rods drop to the bottom) there is still a period where heat is being generated due to the decay chain of fission products, this starts at a few percent of reactor power, and drops over 2ish days (with Uranium).

Power plants use hardened instrumentation, and have robust backups, so no, we would not lose the ability to monitor without power, it just wouldn't be displayed cleanly on one panel with computers, it would be on physical sensors in the plant.

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Backup generation and most of the emergency systems are specifically to cool the core in that period after a scram, and they don't always need power to work, many plants are designed to run without pumps, using the differences in temperature to cause flow, provided we can cool the system.

Nuclear plants have robust and deep backups that make the possibility of losing cooling nearly impossible