I've made well over 10,000 laser and waterjet cut knives and I've used well over 10 different vendors. I've also worked extensively as a waterjet and laser programmer, estimator, and operator.
The issue is predicability, reliability, and severity of failure. A laser can produce wonderful results - but not reliably. A laser can turn a $1000 plate of knife steel into scrap in minutes. Every shop is eager to tell you about how great the cutting will be, but not so eager to get out their checkbook when they destroy your material. The HAZ is the least of your concerns. When the laser cutting goes south, it is extremely difficult to salvage your parts because the failure is so severe - often the parts will be welded into the sheet, have huge globs of rock-hard metal fused to the backside, and/or have lost features. Once a laser loses the cut, it often never recovers - which means if the cut goes south on part number 9, then all subsequent parts will be trash. Often the operator isn't watching your job, and even if they are, it can be difficult to tell that the cut is bad. The right way for the operator to handle these jobs is to (a) cut test parts and tweak the laser settings, and (b) stop the machine every 10 parts to evaluate.
Yes, it is common to see a burr ("dross") on the backside of laser-cut titanium, aluminum, and even stainless steel. In theory, a sufficiently powerful laser can be adjusted to cut cleanly, but it can take hours to dial in the settings. Most shops don't want to spend that time, so they just run the machine and allow the dross.
Waterjet, on the other hand, is much more predicable and reliable. Bad cuts are much easier to remedy. If the cut goes south (often due to a glitch with the abrasive feed), it will usually recover on the next pierce. The machine runs slower, which means the operator has a bigger window to catch problems and remedy.
You definitely want to (a) clearly and unambiguously communicate your requirements, and (b) clarify what will happen if your requirements aren't met. Make sure your vendor knows their money is on the line.