Let's start with year numbering:
So, the year after 1BC is 1AD. Huh? The absence of a year zero is confusing and highly annoying for calculation purposes. But you have to realize that this system was developed before the concept of zero was widespread in Europe. Recent modifications such as ISO 8601 had to fix this to allow for a year zero and negative dates.
The start date is also quite arbitrary. Things like the Unix Epoch and the winter solstice are far more efficient at keeping track of time.
Moving on to how leap years days are handled in the Gregorian calendar:
Added at the end of February. Can we say awkward? And then to top that, the obfuscating algorithm to omit leap days every 100 but not 400 years is not only annoying, but also not all that accurate as this will make the calendar be off by a day in only a few millennia. Calendar systems like the terran computational calendar omit leap days every 128 years which is slightly more simplistic and extremely accurate.
The Gregorian calendar, or in actuality the Julian calendar, are fine and are quite appropriate for our society. They are not inconsistent and are quite rightly tied to the seasons. We do not need to seriously revise it, and it has served us well for all this time. It's of no issue.
The reason why I don't believe it's inconsistent is because I grew up with this calendar. The only inconsistency that I see with it is the days in the month of February. It does not make sense how only one month falls short by more than three days. But with that said, it still seems very consistant to me.