Overheating issue

My vehicle had been running decently lately, however when taking it on trips longer than 1.5 hours, it would begin to run hot. The temp gauge on the dash would go up near the red zone, but then settle back to normal. It would do this several times for the next hour of the trip, but never actually overheat.

I was putting a few ounces of coolant mix into the radiator regularly to top it off, since I have a leak in my heater core that was dripping inside the vehicle.

We decided to investigate whether the issue was the thermostat acting up, and found out that the previous owner had removed it, and I have been driving it like that since I bought it a few years ago.

We put in a new thermostat and bypassed the heater core so I wouldn't lose coolant anymore. At this point, I road tested it, and it got hot and stayed hot after only 10 minutes of driving.

I was used to seeing coolant running left to right across the top of the radiator when I had been topping it off regularly. Also, there was a brown sludgy residue in the the coolant for a long time, and it often sludge up the overflow jug. I'm not sure what this sludge was, but our guess is that the previous owner had a leak at one point and used one of those leak-sealing tablets that dissolves into the coolant (supposedly).

Anyway, the coolant hadn't been running across the top of the radiator for a few weeks before we touched anything, so, fearing that the sludge had blocked up the radiator, we replaced it. When doing so, we completely flushed the whole system (in the block too) with a garden hose, and saw only a minimal amount of the sludge in the exiting water.

We also replaced the fan with a new one, since we thought maybe the clutch was bad on the old one and it just wasn't blowing enough air at temperature. We also checked that the secondary electric fan comes on when the A/C is on, which it does with no issues.

So with a bypassed heater core, new thermostat, new radiator, new fan clutch, complete system flush, and new coolant mix, I took it out for another road test. This time it lasted 15 minutes before actually fully overheating and blowing steam.

I don't have a lower fan shroud, but I know that's not the issue here. We don't know what to do next. Any takers?