Tinkering With The Calendar

Ever since we began counting days we seem to have failed to get a full grasp on the calendar, using the path of the moon and adding the odd days here and there before losing ten of them in the 16th Century in an attempt to rewrite the whole calendar and then plonking an extra day on the end of February every four years to try and bring everything back together again.
What we have now is a year of 52 weeks split into months of 4 weeks 3 days, or 4 weeks 2 days one or 4 weeks exactly or 4 weeks and 1 day every four years which seems overly complicated.
Step forward the The International Fixed Calendar which has been gathering dust since it was first proposed in 1902.
I have been thinking about this and it does have some merit but needs a little tweaking.
This other calendar makes a month exactly 4 weeks or 28 days long and then tags on another month and have 13 months in a year.
What about the leap year i hear you ask but here is where it needs tweaking. Under my system the year would be 364 days long so to stop us falling out of sync with a planet taking 365.25 to complete a full lap of the Sun, we keep the leap year every four years to take care of the 4 x .25's and each year we have a non-day day, like an extra bank holiday so a day which just doesn't count.
It makes sense to me to make it New Years Day and start the year with the 1st January the following day, it already has a name and it is already a Bank Holiday so no messing about with when to slide the extra free day in.
The downside is that with a 4 week, 30 day month, some well known dates will get shifted around so for example my birthday is 1 April which under the proper calendar is Day 91 but under the new system Day 91 would become April 7th or Christmas Day which is Day 359 would become 23rd of the newly named month and New Years Eve the 28th of the new month. 
Every first would always be a Monday, every 28th would be a Sunday and every payday would always be a Friday and the 8/9, 14/15 and 21/22 of each month would be a weekend.
I'm sure there is something i have overlooked but to save any more tinkering with the calendar, perhaps one big tinkle to make each week 7 days long, each month 4 weeks, slide in an extra month to make a year 13 months long with 364 'real' days in the year and a 'free day' thrown in where we get a breather before doing it all over again.
I'm sure calendar makers may kick up a bit but they would be drowned out by all those people born on 29th February who will now have a day of their own to celebrate on each year, March 4th.