It could be the smaller of the three power tubes (two very large tubes, and one smaller one). That small one regulates the series field winding on the motor. If the field has current, the start relay stays pulled in when you press the start button. The fact that you heard something means the forward/reverse lever interlock is working as it should. The other interlock is the gearbox lock which is energized by the same tube. So likely you'll be unable to change the gearbox settings until you get a field current going.
From what I've seen fixing our Monarchs at work, it's usually the small gas filled triode that goes bad, not the two big tubes. Those big guys are pricey ($1400 each)!!! The small tube is available from Monarch directly and cost $488 each (I just ordered three for work last week - including spares). They can be gotten for less online if you look around.
Unlike old TV and radio tubes, the power tubes glow blue when they're working right. This was the sign of a bad tube in the old TV/radio days, but these tubes are good if they glow like that.
Also, there is a 3A fuse inside (in the tube tray) that is for the field circuit. If this fuse is blown, the start relay when click in when pressed, but drop back out again when released. You'll know the start relay is engaged when the light on the start button turns off and stays off.
In your case, your lathe is an earlier model that has extra tubes for rectifiers instead of silicon diodes like the ones we have (1967 vintage). The tubes your lathe has for this function are are just regular radio/TV tubes and can be bought as NOS (figure on no more than $20-$30 each).
I say buy the old girl. I'm well versed in the circuitry inside and can advise you how to fix it. It does pay to VERY cautious. 600+ VDC is inside....
Your other option is just to loose all of the old tube speed controls and retrofit it for VFD. There's tons of info online about doing this. But, I'm nostalgic and like to keep the old tube circuits going...
When you look inside for the first time, the maze of wires and relays will make you think twice about repairing it, but it's really not that bad...
I love the Monarch 10EE and wish I had one myself. I'm still mopping up drool from DMS's score a week or so ago. Until I score one, I'll just have to machine vicariously through you lucky guys that have your own.
John