Electronic – What kind of oscillator should I use for a binary clock

I'll not design the whole thing for you, but plant some seeds.

I could see the foundation of this project being ripple counters. Ripple counters 'count' the number of clocks coming in with a binary output representation. They're available (at least they used to be - do people still use TTL and CMOS discrete logic ICs?) in a variety of sizes - 4 bit, 8 bit, etc.

You could start by using a representation of the input AC frequency to generate the 'tick' signal using a counter, and some gates to detect when the AC peak count reaches the expected line frequency.

You then have multiple ripple counters to count those ticks and represent the hours, minutes and seconds (as well as controlling the LEDs), and some more TTL logic to control when to increment the minute counter/reset the second counter, when to increment the hour counter/reset the minutes and seconds, and when to reset all of them to zero (i.e. midnight).

Once you come up with an idea of how many gates you'll actually need, you'll understand why these things are micro based these days.

It sounds like your oscillator is designed to drive a CMOS load. With the load you gave it, it has to produce 66 or 100 mA to reach a high voltage (66 mA for 3.3 V, 100 mA for 5 V), and it's probably not designed to do that.

One option is to terminate with the equivalent of 100 ohms to Vcc/2:

^{simulate this circuit – Schematic created using CircuitLab}

This reduces the output current required from the source by half.

Even better, if you're able to do it, is route the signal past the two loads in sequence and terminate only once. Since you're able to produce 100 ohm tracks, use that geometry for the whole path:

^{simulate this circuit}

This reduces the current demanded from the source by half again.

Of course it's also possible to add a buffer amplifier that can drive the relatively low impedance of your load, but that adds components you might not have budget or space for.