A drop ceiling can help a lot. Nailing plywood will not help as much - the reason is the vibration will travel through the directly connected plywood. But a drop ceiling suspended by wires will not transfer the vibrations through the wires.

Most sound proofed music rooms/studios I've seen use at least foam-like dampening material. on the walls and ceiling.

Simpler fix for your two machines unless you plan to extensively expand your gameroom in the future.

Plug the center channel speakers in the bottom cabinets with a circular piece of soft foam used for couch cushions. Non-obtrusive, removable, and non-damaging.

Sound will be reduced by roughly 50%.
Vibrating due to construction is another matter. That is low frequency.

The long term solution can be solved by visiting a music studio (for a visual reference) and purchasing hexagonal tile panel soundproofing for the walls and ceiling overlaid with regular building materials to hide their surfaces. Today, there are also now special carpet options to provide soundproofing integrated into floors.

you know what works really good. What a lot of bands do for practice rooms is take Egg cartons (preferably the cardboard ones) and staple them to the walls. Keep them close together and you would be surprised how deading the sound is. Works really well and sounds like a recording studio. I saved up a bunch of them just for that purpose.

If you have more than 10+ machines in one room running simultaneously, a separate independent room, preferably a basement is required. It is very diificult to moderate the overall volume and activity, except with studio soundproofing. It will be loud, not just due to machines. I have lived in duplexes (and very large apartments, townhouses, condos, etc), but my neighbors liked pinball. It was an evaluated prerequisite.

The only thing that stops sound transfer is an air gap. Foam and carpet and other stuff stuck to the walls or ceiling deaden the reverb and make the sound inside the room better, but don't stop the transfer through the walls. I have built three music studios into houses with 90%+ sound proofing by building a room within a room, including opposite-opening double doors. Normal frame separated from the surrounding frame lined with high density dry-wall will do the trick. Of course you are going to lose space in your room, but it will work. I guess maybe too expensive though, as it is basically a full room build. Nothing else comes close in terms of performance though.

When your done sound proofing, for a added level of sound protection you could also install my headphone kits. Hit the mute button and crank the headphones louder then you crank the speakers. The clicking from the game will be minimal noise compared to the level your blaring in your head
That is if your worried about the extra noise you create playing

That's actually for acoustic reasons... not soundproofing.

I recently replaced the doors to my basement with solid doors. They used to be hollow indoor doors. It makes a big difference.

One thing you may not be able to avoid though is the duct work carrying the sounds from the pinballs to the floor above.

TimeBandit is 100% right. This idea of visiting music studios and visually copying what's on their walls is a bad suggestion. Music studios apply acoustic treatments to improve sound quality within a space...They use other techniques to stop sound transfer.

Aside from independently isolating your room (which isn't possible in your case) you need to increase surface mass on your ceilings and or wall (sometimes folks will apply a rubber matt layer between wood or carpet flooring and a floor) or add a second layer of drywall to your ceiling...but you'll have to use things like green glue, etc to help stop transfer between the ceiling and the joists. You can also add dense insulation in the joists (and then apply outer layerings), but you still have to leave an air gap. Drop ceiling will also help.

When I did my game room, I put a drop ceiling in. Not particularly for sound but asthetic reasons. I also put insulation in the ceiling. I am amazed how the sound does not travel above. The game room is below my kitchen which has a tile floor and a dining room with a wood floor. There is alsmost no sound getting through. I think it's also important to put good ceiling tiles as well.

If you decide to add acoustic material such as foam or egg crates, make sure whatever you install is fire rated or treated. Non rated foam materials or similar combustibles burn like gasoline.

I think I can help here. I'm building a theater that sits right below my master bedroom, so soundproofing is high on my list. I'll warn you though, it's expensive.

Think of soundproofing like a submarine. It's pretty much all or nothing. Cutting corners is like building a submarine and then punching a hole in the door. You have to treat everything and I mean everything. Electrical outlets, light cans, HVAC duct work, etc.

I have a build thread going on my basement, but here is the nuts and bolts of the theater.

1. Walls are built 1" shorter than the joists.
2. Walls are decoupled from the joists with IB-3 Clips
3. Walls and Ceilings are insulated with R-19.
4. IB-1 Clips are used to decouple the drywall from the studs
5. Backer boxes are built for any recessed lighting
6. All outlets get putty pads to stop noise transmission
7. HVAC is addressed in a number of different ways (Dead vents, mini split system, zoned system, etc)
8. Double 5/8" Type-X Drywall is hung separated by a layer of Green Glue
9. All joints are caulked with acoustical sealant
10. A solid core door is hung that is typically reinforced with layers of MDF
11. Air tight door seals are installed

My room should have an STC somewhere around 65 - 70.

Your situation might be a little easier since you're not going to be trying to protect against low frequency like I need to in the theater. Unless you have your pins hooked up to subs. The lower the frequency, the more robust a solution you'll need.

The best thing you can do, is contact Ted White with The Soundproofing Company and explain what you want to do. It's one of the foremost experts on soundproofing. His help has been instrumental. He'll be able to recommend a solution based on your needs.

https://www.soundproofingcompany.com/