I don't use LTSpice, but I assume the parts conform to reality. The 74xx74 flip flops have both asychronous reset and preset. Since the parts themselves do not have any other reset functionality, the simulator cannot provide such a function - it's up to you. The simulator is simulating part behavior, not logic behavior, so a built-in reset would not faithfully simulate the circuit.
The circuit you want is called a POR (Power On Reset), and is usually provided by something like
simulate this circuit – Schematic created using CircuitLab
where the inverter is a Schmitt trigger such as the 74HC14. This works in a simulator since all simulators default to zero volts across a cap at startup. For driving 74xx logic, of course, a second inverter is required since TTL almost always uses negative logic for reset and preset functions (with a few exceptions such as the 7490 and 74173).
In simulation, this can be replaced with a pulse generator which produces a (usually) low pulse for a few 10s of nanoseconds after simulation begins.
And yes, you need to be careful about the reset function interacting with logic behavior while it's active, but you need to do this anyways in a real circuit. Fortunately, asynchronous resets (and presets) in most logic chips override all other inputs, so there is usually not much need to worry.