The original setup has 2 wires connected to the capacitor on the OE box. I want to replace that box with a smaller capacitor.
You said
If you just want to ring the bell without it being
connected to a phone line or the other functionality of the phone then you need to generate about 75VAC at 25Hz and connect that to the bell.
So I took that to mean to connect the wires that would normally go to the OE capacitor to your recommendation.
The +/- wires from the ringer go to the Arduino to activate the ringer.
Then you need to generate 60Hz. However that surprises me as I have 40+ years experience in the telecommunications industry and I've only ever known of bells that require around 20 to 25 Hz roughly. Are you sure it needs 60Hz?
Okay, I'm just brainstorming now, so bear with me.
If I was gong to do this, I'd get a small step down transformer, say 115VAC to 9VAC and experiment with driving the secondary with a stepper motor driver at 25 Hz. What I would expect this to do is generate a 75-ish volt signal on the primary that you could use to ring the bell.
So anybody else jump in with ideas. My Uncle AC (great name for an electrician, right?) demonstrated to me when I was 10 that you could get a shock from a transformer and a 6V lantern battery. I don't have a motor shield or I would try it. Probably just a MOSFET would work but I'm trying for something OP can actually build.
Edit: Actually you might be Abe to get a relay to chatter at 25Hz to drive the transformer.
I can add the following. The ring signal varied from country to country but they were all close. Don't hold me to this but pretty sure the exchange limited the ring current to 30 mA. Been plenty of years since growing up with rotary dial phones. Using an Arduino or similar uC I think you can drive a small transformer using an H Bridge.
Along the lines of EmilyJane's thinking a Google of "old telephone ringer using h bridge circuit" brings up a pile of circuits. Including a few multivibrator circuits driving a center tapped transformer.
The capacitor was only there to block the DC component on the line. As a kid (1950s) more than once I got bit by a ring signal on the phone line. Had a fascination with telephone lines. My guess is likely 15 mA would work fine so it's not like much current is needed. Still have a few old rotary dial phones in the basement.
Perry, Emily, and Ron know what they're talking about, but I thought I'd just add that you can BUY "telephone ring generator modules" that will create the correct signal for you. They seem to be a bit pricy ($30 for this one), but by the time you start talking about motor drivers and step-up transformers, it might be an easier solution.