Yes, batteries definitely need replacing after a year or so, maybe 2 if the original battery was very good quality. Don't go with the cheapest eBay batteries either; Monoprice seems to be good if you buy directly from them. I've had my phone for a long time, partly because of the bad phone options with diminishing key features recently (SD card, removable battery). I've replaced the battery every year, and I can get down to 1% battery. When it runs down faster and can't make it that low, it's time to replace the battery.

Now, why this happens in a brand new phone like the Pixel is another matter...