have u checked with meter / scope the reset voltage on the Si4737 ?
your right , the 2k r6 does go to gnd, so if u have 5v on the rst in pin , then it will work as u say. is the rst making 5 volts, and for the right timing.
I've just had a look at the data sheet, the ltc4313, do you have the -1,-2 or -3 version.
fig 5 shows your system, but its with the -1 part, but as u say the i2c seems to work,
data sheets are confusing, ah.
I'd go back to first comment, "whats the eval board and software do". use that as your first example to work off.