Do I have the right conceptual understanding of the North Atlantic Current and its effect on climate?

I have worked with several mathematics equations related to climate change in a math course I have taken. However, I want to make sure that what I have gleaned from this math aligns with what is actually happening (simplified to some degree since I am interested in understanding it conceptually, not to get bogged down in the detailed physics).

My understanding is that both salinity and temperature are the main drivers of thermohaline circulation. More saline water is more dense than less saline water/fresh water, and colder water is more dense than warmer water (for the most part, except right before ice formation). In the North Atlantic current, the surface flow is largely made of warm and saline water. Even though the water near the Greenland-Norwegian Sea and the Labrador Sea is less saline, the water in these regions is still highly dense due to the low temperature and (maybe?) ice formation keeps it fairly saline (i.e., when ice forms it pushes out the salt and keeps the freshwater).

This is where I lack some confidence in my knowledge or get confused, and perhaps you can help me fill in some gaps:

• I think the North Atlantic Current is slowing due to sea ice melting and making the water less saline (i.e., less dense), as well as the warming temperatures near the poles making the water less dense. This means the water will not as readily "drop" down to deeper levels of the ocean (also is there a better way to describe this?). Without deep water formation the conveyor belt isn't being "pulled down" to these deeper levels (and not allowing more water to rush toward the poles due to the lack of water being displaced to deeper levels), which has an effect on slowing the circulation in general.
• Based on the math equations I have seen, I also see that the linear approximation for the flow rate (i.e., Stommel's two box model) assumes that temperature difference and salinity difference between the polar and equatorial regions of the North Atlantic current are working in opposing directions. What I mean by this is that the warm water in the equatorial regions of the North Atlantic tend to "pull up" toward the colder water in the polar regions due to this (horizontal?) temperature gradient. However, as the poles warm, this temperature gradient weakens and slows the rate of surface flow in the North Atlantic (which can cool parts of Europe). For salinity, I know that the equatorial regions are more saline than the polar regions. My understanding is that the freshwater melting from sea ice in the polar regions flows toward the more saline water (not sure how to explain this but it reminds me of osmosis but I know it's not). This, from my understanding, further decreases the rate of (surface?) flow because this is again working in the opposite direction.
• I realize my understandings of impacting the temperature gradient and salinity gradient and its impact on climate are mostly within the surface flow. How could I fit in the deep water flow into this?

I just wanted to write everything I have synthesized from readings so far. Please let me know anything I am missing, ideas I have wrong, etc. Thank you!