Fats and greases are perfectly natural. No argument there. But fats and greases are also a barrier preventing the hide glue (another natural component) from binding mechanically to wood (yet another natural component).
If you have even a hint of grease in the sinew, naturally, it will not bind as well as it is capable of. Using wood ash and water creates a weak lye solution that will react with fat to turn it into a natural soap that can be rinsed out. That's about as natural as it gets. All soaps are a variation on that theme, so in a sense, that is all natural, too.
In bowmaking, it is all about tradeoffs. You trade off speed for long term reliability, width versus depth, this versus that, etc. This is just another thing you have to balance for yourself. How risk averse are you to having your bow fail by the sinew delaminating? If it would bother you to put in loads of hours of work and the time/cost of materials just to have it turn into junk at the wrong moment, then trade a little time with Dawn dishwashing liquid wash job on the sinew.