I'm willing to spend substantial amounts of time in my remote market, but I need it to be scheduled ahead of time--I just can't jump on a plane every time a toilet needs to be fixed. 

One one end of the spectrum there is 1st TD paper with modest loan to value & then there is toilet paper (that's what was created by Wall St. in the bubble).

You can do a cheaper renovation which may save a month or two. or you can do what I did. you can rent it unfinished. at 60% of your normal asking price.There are many people who would love to save a few hundred a monthall they need is a working toilet and a microwave and a small refrigerator. and hot plate and toaster oven.

I've done a number of different things without previous experience by doing some research and getting the right tools (tile backsplash, moving a toilet drain in a concrete basement floor, etc.)

Toilet and tub will not drain, what's wrong with the plumbing?

Fixing a toilet?

I'm quite handy with tools, but never really make a major DIY project like tearing up a wall (but I could replace toilet/sink, put tile flooring, etc) - how important is the ability to fix almost everything in a rental place?

Late at night or at certain levels of inebriation, it's an easier target to hit than the toilet.

The agent told them the market is in the toilet, no one is interested, no one even came by to look.

The flat is worth €1500 per month in rent and is includes 1 bedroom 1 decent sized living room, full bathroom, half bathroom (sink and a toilet) and a kitchen.