There is a reason lead ones are not "just" heavy but also physically much bigger than same capacity lithium one.
What will (probably) happen if we get competitive sodium batteries is that they will phase out lithium from the stationary storage so there won't be that huge demand for lithium for everything. Or possibly they will become "cheap EV's battery"
Sodium cannot improve lithium by replacing it in batteries, for physical chemistry reasons which can't be negotiated.
I don't consider it likely that lithium will become so much more expensive that people will want their second-largest purchase, a car, to be inferior in the way that it must be with sodium batteries.
Rather more likely that cheap and good sodium batteries relieve pressure on lithium supply for the enormous amount of battery storage we need to handle intermittent renewables.
By weight, something around 1% of a lithium battery is lithium.
Sodium is seen as an interesting alternative, because there are some indications that it can make that 99% that makes up the rest of the battery lighter. Even though sodium is ~3.5 times worse as a charge carrier, if it's use allows cutting enough weight from that rest, it can be a winner overall.
For example, aluminium is, for complex reasons, not compatible with every part of a li-ion battery, and so they use copper connectors internally, typically containing by weight at least 10 times as much copper as they contain lithium. Sodium batteries can use aluminum everywhere, which saves on not just cost, but also more in weight than the weight difference between equal molar amounts of sodium and lithium.
This is absolutely not clear. Active lithium is just ~60g per kWh in a battery, or ~1% of the mass of a state of the art LFP battery. Even though sodium weighs 3.5 times more for the same charge, if it's use would allow just a few percent reduction in total weigh, either by reduction of inactive cathode material or some other source, it could easily beat lithium batteries.