Don't get it... how exactly would getting rid of the toss reduce home advantage? Let's say a country is planning to doctor their pitches, why would they change their minds if there's no toss? Stats show that winning the toss doesn't have a significant effect on the outcome of the match either, so what's the point? Not to mention that Joe Root won 4 tosses in the last Ashes, we got smashed in all 4 games.
Want to reduce home advantage? Here's a suggestion - decree that limited overs series must always be played before test series, not after. The best players play both formats, so they'll be nicely warmed up for the main event. Warm ups must be first class matches, touring sides do themselves no favours by playing 15-a-side nonsense two day games. There's no need to mess with test cricket, but if you want it to be the pinnacle of the game then treat it that way, simple as that.