The heat reservoir keeps the temperature constant, so P·V must remain
constant, but why couldn’t P increase? Why must V increase?
The key to the reversible isothermal expansion process for an ideal gas is you have to very slowly intentionally reduce the external pressure while allowing the volume to simultaneously very slowly increase during the heat transfer, so that the product Pv is constant, the temperature of the gas remains constant, and the work done equals the heat transfer in. Per the first law, there is no change in internal energy.
Or why wouldn’t P and V just remain constant? The temperature of the
gas and temperature of the heat source are the same. There is no
reason for heat to flow into the engine
You are misunderstanding the reversible isothermal heat transfer. The temperatures of the gas and heat source cannot be exactly the same, because then there would be no heat transfer. For the Carnot cycle isothermal expansion the heat source (thermal reservoir) temperature is infinitesimally greater than the gas temperature in order for heat transfer to be possible. Then the ideal gas does an infinitesimal of work equal to the infinitesimal amount of heat transferred in, thereby keeping the gas temperature constant. Its the same situation with pressure. For the expansion the external pressure is kept infinitesimally less than the gas pressure.
All of this is why the Carnot cycle is an idealization. All real processes are irreversible. In the limit we say can say they are reversible.
Some descriptions say that P is reduced by external action, for
example, “the weights on top of the piston are removed, so that the
gas can expand.” But if energy is applied to the engine (to remove the
weights) that doesn’t sound like much of an engine. Where did the
energy to remove the weights come from?
Theoretically you can do this with the expenditure of little or to no energy to "remove the weights". Consider the following thought experiment:
Imagine the weights actually consist of a pile of sand placed externally on top of the piston.There is no friction between the sand and the surface it rests on. Now imagine a vertical platform along side the rising piston with multiple holes in it.
We take a single grain of sand and we slide it horizontally into a hole in the platform. The mass of the grain of sand being so small and since it is sliding horizontally (no work against gravity) on a frictionless surface (no friction work) the work required to remove the grain would be infinitely small, certainly much less than our engine output.
When the grain of sand is removed, the frictionless piston moves up a tiny bit and we slide the next grain of sand into the next hole, and so forth. In this thought experiment for each removal of a grain of sand we are decreasing the external pressure and increasing the volume a tiny bit, doing a tiny bit of work while transferring a tiny bit of heat to the gas, all of which results in a process that proceeds infinitely slowly. Which is why such a process would be impractical and is never actually used. It sets an upper limit on the effeciency of all real processes.
Someone once said putting a Carnot engine in your car would give you phenomenal gas mileage, but pedestrians would be passing you by!
Is it just that, no the piston would not actually move, but here is
how much work would be done if you rigged things so that it did?.
Given the above thought experiment, perhaps I have adequately addressed this.
Hope this helped.