In fact, we see our blood pressure as periodic, and feel our arteries pulsating. But our tissues don't see that. What I mean by that, is that tissues get their blood supply ultimately through their capillaries, where the blood pressure and flow is essentially constant throughout the cardiac cycle, as you can see on this chart:
Image from OpenStax. Article on Blood flow and pressure
As for the second part of your question, what would be the consequences of having constant/non-pulsatile pressure? We can only speculate, I don't think it would have any palpable consequences in principle.
The only "real-life" examples include people who are, for instance: on a Cardiopulmonary Bypass during a cardiac surgery, on Extracorporeal membrane oxygenation (ECMO) in the ICU or have a left ventricular assist device (LVAD) for severe heart failure. It's very difficult to infer anything from these situations, because any on these patients are already very sick, and consequences following these procedures are usually due to insufficient oxygenation, thromboembolism and such, which have nothing to do with the constant pressure.