Changing head gaskets is a PITA specifically for your back but it's not high tech difficult. More labor time/grunt than tech.
"Shop quoted a few thousand to repair the head gaskets." Yikes sounds like the shop is busy and doesn't want the job unless they at least get a new fishing boat out of the deal.
A:/ find another shop. The grinding noise is likely a flexplate bad spot with broken teeth. Not a big parts expense but labor intensive. Gotta remove the trans and TC and going by what they charge for shop time...find another shop.
You can easily inspect the teeth by just removing the trans inspection cover and rotating the flexplate with a big screwdriver etc. Might just be a trashed starter bendix drive/gear or combo of both?
If you can find a "reasonable priced" shop that doesn't like fishing, a blown head gasket isn't yet a call for a new motor.