Have an old style double sided kitchen sink that is attached from beneath
the countertop. There are clips of sorts holding it up. There is a small
gap between the basins and the metal trim/casing. The newer basins, from
what I am told, rest on top of the counter.

The faucet leaks at the base when you turn the spout to the right. The
water flows to the gap between the basin and metal casing. The water then
leaks into the cabinet below.

A couple of my plumber friends say I should replace the whole counter,
sinks, and faucets. I really don't want to put that much money into it.
Would rather just fix the leaky faucet and seal the gap between the basin
and metal casing. They don't seem to like that idea at all. I guess it
must be difficult or time consuming to fix these types of leaks.

For now, I just don't use the right sink. Would like to just replace the
faucet, but one plumber said don't bother as it still would leak between the
seams. My guess would be if I replace the faucet, all I would need to do is
caulk up the gap to prevent any residual water from going down.

Is my thinking faulty? Is there a caulking that's safe to use in the
kitchen? Are faucets difficult to replace in an older system? Feel like
I'm fighting an uphill battle with these plumbers.

Wouldn't think of replacing the faucet myself. Seems like they are
impossible to get off.