Electronic – Counter is missing an output signal

With only simple logic gates, you're going to be "putting a crazy amount of them" on the four MSBs. Optimizing your logic can help reduce the number of gates (there are two that start with 11, so they can share some gates, for a start).

However, if you were to spend a few pence and invest in a 74xx154 4-to-16 decoder then the 16 possible combinations of MSBs would be presented to you as one of 16 outputs being pulled low. Much neater than lots of gates.

If you can't get a 154 you could use two 74xx138 chips and use the highest MSB bit to choose between them.

Am only going to address your question about reading the count output from a '393 counter. You have two problems to overcome.
This counter is asynchronous, which means that a clock input ripples through the individual flip-flops to its next count state. If you happen to read the count value on-the-fly very soon after a clock transition, you might get a really strange number, neither one less, nor one more than what you'd expect. This rippling occurs very quickly but nevertheless can cause errors.
The other problem occurs when you concatenate counter chips for extended count values. A microcontroller can read in eight bits at-a-time. To read in a count from two or more '393's, the microcontroller must read in eight bits from one chip and save that result into a register, then return and read in eight bits from the next chip. In the interval, the count value can change, leading to the possibility of a wrong result.

^{simulate this circuit – Schematic created using CircuitLab}

One solution to both problems adds a "gate" to clock input. When the microcontroller wants to read the count value, the clock input is disabled, freezing the current count value. Then the microcontroller can read the count value, which is now frozen at some value.

A 74HC590 counter chip is more appropriate. Its counter flip-flops are arranged to make a synchronous counter, having no ripple-count problem. It also incorporates an internal gate. It also provides an internal counter latch that stores the current count value, while letting the counter flip-flops continue counting up. And it provides a tri-state output, so that a micrcontroller can read in its eight-bit value from the same port. This makes it very easy to concatenate these chips if your microcontroller has limited I/O pins.
However, most microcontrollers have versatile internal counters that can give similar results with no external chips.