White solid drain pipe vs black flexible corrugated pipe for a french drain

Your basic plan sounds workable. I assume your crawl space is adjacent to a full basement area? I have a couple of questions. Where does the existing drain tile system empty? Is there a difference in level between existing basement floor and footings of the crawl space? the answers to these questions may change my answer to you. But basically, adding a French drain or perimeter drain tile should help your situation. After your trench is dug, be sure to line it with several inches of gravel/crushed stone. Use perforated pipe with inlet holes or slots on only one half of the surface and be sure the slots are facing up when installed. Wrap the pipe in common landscape cloth before backfilling with crushed stone to prevent the drain pipe from filling with sediment. Lay another layer of cloth over the stone and complete the backfill with clean grave. Not sure how you plan to connect to the basement drain system, but that is another topic. This kind of work is always difficult, especially when working in a crawl space with limited head room. Good luck.

A power trencher, or Ditch Witch, is a fairly easy machine to use. Most are self propelled. You may check some local rental stores as well as HD, as you will probably need a trailer to transport the machine and most rental stores include it in the price.

Before you attempt to use one of these monsters, be absolutely sure to call DIG SAFE @ 1800-digsafe. You must have the utility companies come and certify that there are no buried lines, wires, gas etc. It is the law, so don't forget to get it done. There is no charge for this service. If you don't and hit something, you can be liable for huge costs and fines.

A Ditch Witch usually only digs a narrow path for laying wire or pipe. The path width would be a bit shy for a good drainage system as you will not have enough width to add the proper crushed stone base and surrounding gravel bed etc.

Assuming you are installing wire or pipe, be sure to get some warning tape to install a few inches below the ground grade to warn future excavations that utilities are buried benieth. Good Luck.