The neglect issues with the tecumseh mostly comes from the carb. They didnt have fuel taps so leaking needle and seat is an issue.
They hate ethanol as do all mowers. Water in the carb, most ive fixed have had it.
If they havent been run dry on oil and the air filter has done its job, usually a carb clean out will fix them.
Used starters do OK on ebay, $10-15 seems to the the going rate.
Coils do OK too, i find that strange as i have never had one here thats had no spark.