how do game companies maintain public servers financially? without monthly service payments?
Thats a big and broad question
You need to a) have a notion of cost per player b) ensure you factor that in to your back end cost plans c) have efficient and elastic resource utilization d) have good metrics and e) have plans for user acquisition and high value player retention.
To support free-to-play models you need to investigate subscription models (battle pass, season pass) or enough revenue through IAP/micro-transactions that have value to your player base.
Or you can factor in some lifetime costs per player and thats part of the game up-front purchase prices. Games will typically operate services for x years. The better ones also support LAN/peer hosted models so the game can function beyond beyond lifetime of the game platform services.
If you look at highly successful games, for example Fortnite / Genshin impact, the sheer amount of revenue from the paying parts of their player population is far above what is needed to support their backend services and give them considerable financial buffers to operate for years.
At a smaller scale, Amazon GameLift allows to take advantage of multiple servers running on EC2 Linux spot instances which you can elastically scale meaning you save on your hosting costs but also allowing elastic scaling during peak demand.
Hopefully others will have input here.