You can install a tankless water heater fairly easily. @$2200 is likely normal parts and labor for your region. But if you have time ( and can deal with leak) start working the phone to get estimates. Many home repairs have a great deal to do with psychology and salesmanship. When contractors show up you need to sell the idea that what everyone is looking at is a nuance but not an emergency. The leak everyone is seeing is something you will deal with until boiler just gives up. The more you plant seeds of simplicity, through a little research, it knocks the edge of of aggressive time line and pricing from plumbing contractor. Removal of old, heavy, bulky water heater is not for amateurs or faint of heart. Everything, everyone needs to be in place, couple of Fabio sized guys to get it into a truck that day. The tankless is the size of a suitcase depending on the flow level you need, more gallons per minute, bigger suitcase. Your homework is key, what are your existing conditions, electric, gas and the rest is the existing space or cavity where water heater is currently. Gallons per minute is really the only concern. How much hot water you use..In my experience I was wary of having someone washing dishes and both showers going simultaneously. It's happened but rarely. Are we doing wash and the machine is filling, 2 people need to shower and dirty dishes in the sink all create a hot water demand emergency.. Nah.
And the diagram really is revealing... If you have gas you need to extend the pipe from old water heater, electric is self explanatory also and it's incoming cold water and outgoing hot water. I have used PEX pipe for a few years with zero issues.
If you talk to your possible plumber with some knowledge and buzz words everything can go smooth.
We want to install it here, obvious, and just do PEX extensions to get it up and running, we don't really need to spend time and money creating another broom closet. Tankless is a fifth of the size , just looks like a banana box size thing mounted on the wall.