I spent a lot of time with this circuit as well as another that changed out Lars' simple DAC for a two stage DAC system and used a TDC7200 for the TIC. But prior to going that route, I had a number of issues that were pretty basic. Your mileage may vary.
1) Make sure your PPS isn't overdriving the 4046. If it is, and probably will be, use a simple voltage divider or variable resistor to bring it down below the 5V (if using 5V logic) and then use a 100 ohm resistor to help reduce the ringing. There is usually a big spike on the 1PPS signals a that slew from zero to 5V pretty quickly. That helped a lot for me.
2) Same thing with your 10Mhz input. Square it up if needed using a two-transistor or other circuit from Wentzel. Go here
https://wenzel.com/library/ and search for squaring circuits. I've always used the two-transistor as it was called. You can also look at the TAPR TADD or TADD-2 or TADDD-2, can't remember all the 'D's in it, ha, input schematic. Again. make sure it isn't under or overdriving your logic. This is on the return signal coming from the oscillator.
3) On circuits like Lars that use saw-tooth correction, again, make sure you have the logic levels correctly in phase and aren't overdriving the Arduino. I had to use a voltage divider again.
4) Breadboards that have been used a few times start to loosen-up. After a lot of prompting, I changed most of my little colored jumpers to single strande, tinned, 20 gauge wire. I then doubled-up the key connections. This took a lot of little jumps out.
5) One of the tests it to put it in hold so that initially the TIC is counting around 400 and then watch the output. It should smoothly go or or down with the tic_max and tic_min close to tic_average. ( I think Lars exposes these fields). You can see the jumps caused by poor PPS conditioning, poor Oscillator return, etc pretty easily. Bang on the table a little to see if it jumps. Better yet, if you don't have a PCB, get a few of the solder-type breadboards from Amazon, EBay, Ali, etc and use them. I think Lars circuit costs less than $20 so if you solder it up and need to throw it away a few times, so what. It's easier than fighting breadboard noise and ground loops.
6) If still using a breadboard, take a 20gauge wire, strip it off and slip it down the center channel. Then mark where you need grounds, solder little wires to it, pull the chips, put the wire in the channel and use that as a single point of ground.
7) Alternatively, take a piece of PCB stock and pull the stickum off the bottom of the breadboard on the ground channels. Mark and solder the channels to the PCB. You might need to put little solder bumps on the PCB to lift the channels up so you can keep the stickum on the rest of the breadboard. You can do this for the ground runs from the bus lines as well. Then put the breadboard back onto the channels for the rest of the circuitry. You can also do this with a solder type board. The large ground plane will help while maintaining flexibility.
Use a good oscillator or it really isn't going to work well. Something that floats all over the place is just never going to do much for you. The GPS and TC can only do so much. I used an HP 10811 and X72 Rubidium successfully and it took just a few hours to get the first lock and then a couple of weeks to keep it locked.
9) You need bypass caps everywhere. Especially diagonally across the 4046, HC390 and any other chips. The diagonal was a great idea from a friend.
10) back to the oscillator return. Use an isolation transformer like an MCL T1-6T. You don't ground the input side to anything. You can bend the pins on one side and stick it down in a breadboad (if using one). Make sure the input ground isn't grounding to the rest of the circuitry.
11) Power supply. I had one for the GPS, another for the oscillator, another from the PC USB driving the Arduino, etc. This is a bad idea and I had ground loops all over the place. Some GPS have a 5v output but you need to minimize the number of supplies. The isolation transformer helps with the oscillator because you need a ground to the EFC and if you have a long wire coming back from the oscillator output, that can cause a ground loop.
12) Wrap the oscillator in styrofoam to keep air currents from causing sudden shifts.
Hope it helps.
Jerry