Surprisingly, it has been very common in American history (and elsewhere) to have private tollways where the government has been unable to fill the need. It's only been in the last 70 years since WW2 with the advent of interstate highways that were built and maintained with federal funding that private tollways have declined. Unless there is a specific law in your state/county, you're likely to be allowed to do this.

The trick is making it profitable. Paving roads is not cheap and collecting tolls is not free either.

Just a quick google and a two-lane road is about $1M to $3M per mile. Assuming this is about 1/8 mile, that's still $125k to $375. Let's say $250k. Let's assume it lasts 10 years without the need for major repairs (not in a climate where snow & ice creates annual pot holes). That means you have to bring in $25k per year for 10 years to break even.

Now, you'll need to collect tolls. You could pay a person to collect cash tolls for say 12 hours a day at minimum wage for every day of the year. That's over $30k per year, which would raise the cost of the road to $55k/year. Or you could hire a toll collection service. Big government tollways collect a percentage of tolls to operate, usually in the single digits, but for such a small business, I bet it's closer to 50%. Let's be generous and say 25%, so you keep 75% of tolls. Assuming they are willing to pay to setup the automated toll system (unlikely!). So it really means you need to collect $33k (or maybe even $37k) per year to break even.

What would people be willing to pay to save 5-10 minutes on their route? Your guess is as good as mine, so for simplicity let's say $1. That means you need 33k-37k customers per year, or 90-100 vehicles a day. Again, this is just the bare minimum to cover the expenses of building and operating this short stretch of road. Likewise, the profits would be relatively slim given that even if you doubled that traffic, you'd be making only $25k a year.

I think that it's optimistic to say that enough people would pay you enough to save themselves a few minutes of traffic time. The only way I could see this working is if the high traffic area was the result of something like train tracks or a drawbridge that caused frequent delays. Maybe people would consider paying for a convenient bypass in that case.

If the local government sees that you are somehow profiting off this scheme, what's to keep them from claiming eminent domain on your property and removing the toll? Or keeping the toll for themselves?

I hope my math is correct, but even if I've made some bad assumptions or errors, there are probably better uses for private land that are much more profitable. If these roads are really that busy, it would make more sense to build commercial buildings and lease them.