Q: lately my van often doesn't start on the 1st, 2nd, or thirtieth try.

Under the hood, the engine advances slightly with each click. During this symptom the starter draws about 500 amps. Boosting had no effect. The starter was replaced. I drove off. Minutes later it failed in the same way. My mechanic then swapped two identical (high power) relays in the fuse panel. The van started right away. A couple of days later the exact same symptom happened and left me stranded. I removed the same relays (one of them appeared hotter than the other) and let them cool off in my hands and reinstalled them in the same location (no swap). After a few more (20ish) tries the motor cranked normally and started. Stranded time was about 15 minutes. It should be noted that the underhood temperature is usually high (but not every time) when this happens. When the starter is "engaged and not cranking" the dash lights do NOT dim. Any ideas? Thank you for your time.

My car has 265000 miles.
My car has an automatic transmission.