You aren't required to own specific model or have specific feature in your TV. Newer online multiplayer games may take advantage of 120hz refresh rate in supported TVs. They will feel little smoother to controller but unless you are thinking about buying new TV for $1000+ this isn't something you should worry about.

Use what you have it's fine 99% of games will aim to play at 60fps anyway so standard 60hz refresh rate is perfectly fine. Console will adjust to resolution if your TV as well.

Just focus on console and getting games cheaper if you are trying to cut cost.