Thomas/Carl,
Oops. Yeah. Wiring cock-up, by the looks.
You've got a 110VDC power on/off switch running through a 3.3V Tiny - might be toast. Might be lucky and it's only that pin that's toast, in which case you can play with the source and get another pin to do the same function, or use a different function that controls a logic pin - like the 'cool' pin.
The SPIN function is a logic switch, meant to be wired to a device that supports logic - 3.3, 5, 12v, it doesn't matter as it's the other end that has to see the rise in the level to determine the change of state. I often send a 3.3V logic to a 5V receiver - such as an Arduino. It's input at the 5V end, so as long as the V rises enough from the Tiny to trigger the logic, all good. Doing it the other way round, not good.
That 'controller' is a transformer from 110vAC to 110VDC but they have built in a PWM speed controller. The 'speed controller' is a potentiometer, it can be wired to a Mach3 controller, and the TinyG!
You can achieve what you want.
Check the speed controller pins for voltage = it shouldn't have any (you never know with these Chinese boards... I've seen all sorts). PWM needs to have a signal, ground and +xV. If the signal is happy with ~3V, you're good to go. if not, you will need to shift it to use the PWM on the TinyG. You will also need to provide it with voltage and ground
However, the Tiny is a fragile little beast and I would use voltage and level shifters, as well as a diode to absorb any spikes. It's a small amount of money compared to replacing the TinyG.
https://github.com/synthetos/TinyG/wiki/TinyG-Configuration-for-Firmware-Version-0.97#pwm-group-pulse-width-modulation
I use cheap adjustable voltage levellers. You can get them from Sparkfun or Ebay for cents. I also use signal level shifters, again from Sparkfun or EBay.
At the least to control the spindle power on/off you need a relay that can handle NO current of whatever amps that thing draws - and triggers with a small DC voltage (<5V) - you may get a car relay to trigger that low...
Personally, I would look at something like this rather than that 'speed controller' which you have no idea of the pedigree: https://www.inventables.com/technologies/spindle-speed-controller
And read this: http://blog.inventables.com/2014/06/shapeoko-upgrade-quite-cut-spindle-with.html
As to power relays, I've literally just set one up on the TinyG to control my AC Kress 1050. I used one from WeMos, designed for their 3.3V ESP8266. I use it in NO mode. It's already set up to receive a 3.3V logic signal.
https://wiki.wemos.cc/products:d1_mini_shields:relay_shield
I'm on 220V, but it can handle NO (Normally Open - it's powered-off state. In NO mode, when power to the relay goes off, the relayed power connection is broken. In NC - normally closed - mode, when the relay power goes away, the relayed power switches on - used in some situations, not in CNC.) current of 5A(250VAC/30VDC), 10A(125VAC), MAX:1250VA/150W - I guess it depends what side of the transformer the power switch is on. The spindle is rated to 500W at 110VDC. If that transformer is pulling more than 5A AC I'd be surprised, but I know nothing of your North American wiring standards. 10A is the max here in 220V land, but that gives you a heck of a powerful device on the end - like a 3500W heater or 2500W electric hammer :)
If you're not sure, give me more info and I can work something up. It's a bit like working in the dark here...