So my guess would be it will make no difference. In addition the problem might be the wifi in the camera since it makes up 1/2 the connection.
The best solution as recommended above is some kind of wired solution.
Now maybe you can get better wifi, that router is extremely old, so maybe a slightly newer piece of garbage router would help. You will not see any difference with some high end router as far as coverage goes, as you guessed it is all for speed.
Unlike most modern routers that old linksys does not transmit at the full legal power. At least I don't think it does there are many many hardware revisions of that router. A newer device would likely transmit at the full radio power. Next that linksys only functions on 2.4g. Maybe 5g will work better if you get a router that can do both 2.4 and 5. In general 5 tends to have less coverage but you never really know when you are talking about the signals going through a wall. You cameras would also have to support 5g wifi.
If you really wanted to try this buy pretty much the cheapest thing you can find. A used router from the garage sale for $5 would likely be fine. All you really care about is radio transmit power and any router from a better know vendor made in say the last 5-7yrs likely transmits at the full legal power.