That was always my habit. I would use stop when processing 4x5 sheet film, though.
Cold tap water is generally fine for all uses. The exceptions are when the tap water is of relatively poor quality and hot tap water which too often contains extra disolved oxygen and "grunge" from the old water heater. Old buildings and some older neighborhoods with heavily corroded supply pipe may introduce some undesirable additions to the water. Under such conditions, using heavily filtered water or distilled water is called for. One apartment that I lived in for a while had issues that forced me to use distilled water for a final short wash and for the Photoflo. Other than that, tap water was always fine.
I did take on the habit of bringing the water used for mixing developers to nearly a simmer on the stove before cooling it to the desired "hot" water temp for mixing. This forces out the bulk of the dissolved oxygen increasing the storage life of the mix.
DO should be greater in cold water, but is perhaps not much of an issue in developer life, nor is a bit of air left in the bottle, the big problem is that caps leak air, and unless you put Saran Wrap under a screw top, or use hard to find ground glass stoppers fitted to the bottles, caps breathe air. You can experiment for yourself. Air is less than 20% O2, and probably close to 100,000 x the DO of tap water, perhaps the same for water with algae.
I have seen much longer storage life in terms of obvious oxidation in stored developers. Cheap plastic breathes and those accordion bottles they used to sell just increased the surface area to breathe. And yes, I used to sell bottles and caps and did DO and pH research for an EPA river project. ;-)
Hot water does have more disolved solids, and I do not care to drink water sitting in the house pipes all night long, so I let it run before I drink it in the morning.
The city has to keep the pH a bit high to keep from dissolving the pipes, low pH will dissolve them quickly. However, the calcite ends up lining the pipes, and almost no one flushes out the hot water tank, which probably cuts its life by 50% as the crap on the bottom makes the bottom of the tank heat up more to heat the water.
If you use chemistry at a fairly good rate, much of this remains theory, except for the Saran Wrap.
If I had it, I would use distilled water for developers, and I do use it for final rinse-- seemed to be less important in MF and larger films than 35mm.
And I agree with Tom, I use water in place of acid stop for films, but acid stop for paper, color and B&W. I believe in hardnerer in fix for films, but not papers.
Biggest problems are probably getting the right exposure, development times, scratches in the base or emulsion side and crud in the emulsion. I often wonder what Capa felt when the processor ruined all but a couple of his D Day shots by using very hot solutions causing the emulsion to run off.
Regards, John