Many years ago, I ran across a similar arrangement of a loudspeaker with a paper cone diaphragm attached to a voice-coil that was inserted over the pole piece whose magnetic field was created by an electromagnet, whose two pole pieces surround the voice-coil. The image the OP attached fits this description.
The loudspeaker in question IS NOT an electrostatic speaker. The attached transformer is an audio output transformer that performs the function of impedance matching the output resistance of the vacuum tube audio power amplifier (typically several thousand ohms) to the low impedance of the loudspeaker (typically a few tens of ohms). The electromagnet not only creates the magnetic field against which audio currents in the voice call react against to move the diaphragm, it also serves as an inductive filter for the B+ plate supply. This inductance is large, and there is superimposed on its current an AC ripple component. This AC ripple or "hum" (usually 120 Hz) should be barely audible in a quiet room, even at full audio amplification.
My experience with this technology came as a teenager who "rescued" and restored a discarded console broadcast and shortwave band radio receiver from the trash behind a radio and television repair shop in Denver, CO, sometime in the late 1950s. I used this radio to learn a lot more about electronics back then. I remember using the inductance of the loudspeaker electromagnet to create fat, hissing, DC arcs in air. Why? You may ask. Just for fun, after I accidentally discovered the phenomenon one day while disconnecting one of the two wires feeding current to the electromagnet. Of course, being young and somewhat foolish for a wannabe "sparky" I did this without first removing power to the circuit.
You don't really need a published schematic for these old vacuum tube electronics. Everyone used the same superheterodyne system invented by Edward Howard Armstrong, with only minor variations in how the superhet radio receiver was implemented. Here is a link for a more detailed description:
https://www.microwaves101.com/encyclopedias/superheterodyne-receivers. Make your own schematic by reference to a vacuum tube description of your existing audio amplifier, using a multimeter to trace out the wiring and connection to other components such as resistors, capacitors, and (if the record player has a radio) intermediate frequency (IF) transformers. Lots of fun doing that, and you will learn a lot of practical things if that is your goal. If not, you will still end up with a schematic you can work from. It may not be "pretty print" but hand scrawled schematic diagrams are nevertheless still useful.
If you are truly restoring your record player to "like new" condition, I think that is a waste of time unless it is done as part of a "learning electronics" exercise. If all you need to do is get the record player functional again, while keeping its cabinet and speakers, you are in for a bit of work. Personally, I would keep the turntable and ditch the rest of it. Modern PM (Permanent Magnet) loudspeakers are inexpensive, and solid-state amplifiers are readily available in power ratings from a few watts on up to truly awesome power levels.
OTOH, this is a hobby forum... so build an unregulated, electrolytic capacitor filtered, power supply for the loudspeaker electromagnet (several hundred volts at maybe one hundred milliamperes should work). Then buy an audio output transformer similar to the one already mounted on your loudspeaker and wire it backwards to the existing transformer: the high impedance input winding of this new transformer is connected to the input terminals of the existing transformer, and the low impedance winding is connected to the low-impedance output of your audio amplifier. This effectively removes both transformers from consideration, with some loss of audio fidelity, but those old loudspeakers were pretty awful at audio reproduction anyway (ask our resident expert
@Audioguru who knows far more than I ever will about it). Better yet, if you can gain access without destroying the loudspeaker, remove the existing transformer entirely and connect your audio amplifier output to the voice-coil terminals directly. Do this ONLY if you fail to find another suitable transformer as it is likely the existing transformer has very short but flexible leads connected directly to the voice-coil.
I no longer have the fine, almost antique, console radio with its huge loudspeaker. It disappeared after I graduated high school and joined the Air Force. Mom and Dad were in the process of getting a divorce while I was serving, so I suspect that Dad discarded my radio. No harm, no foul. I went on to bigger and better things.
I should caution you NOT to use your restored record player to play new vinyl records. The steel needle stylus used on those old players was designed to survive the harsh punishment dealt out by the 78 rpm shellac records that were prevalent before World War II. Vinyl records recorded in stereo need an elliptical diamond stylus and only a gram or so of pressure to avoid ripping off the tracks like a snowplow. And there is a protocol, used by those with Golden Ears, for handling vinyl record pressings. My ears are far from golden, if they ever were, so I do just fine listening to digital streaming audio from a compact disc or online via the Internet.