Try it and see if you like it. It's really a question of handling preference, although there are some side issues.
Keep in mind is that (unlike the back tire), the front tire wears most quickly at it's edges. So over time, the profile gets progressively steeper and the steering gets more tippy; that is, it falls into the corner more readily. Starting with a 120/60 or a 120/65 section tire reduces this wear-induced effect.
Further, a lower 60-ish profile tire (often) has a shorter height so it doesn't flex as easily. This increases the severity of the loads transmitted to the wheel rim, suspension and handlebars. This in turn, increases the likelihood of alloy wheel damage, so a 70-series tire will be kinder to magnesium wheels.
Your tires are an integral part of your suspension so when you go from a 60 to a less-stiff 70 tire, the resultant handling change will probably benefit from front suspension setting modifications. A change to front compression damping may be in order to offset the less-stiff 70-series tire. Also, turn-in is slower with the 120/70 but this is easily compensated for by adjusting rear ride height
Also, an increase in the outside diameter of the tire will cause your already inaccurately high-reading speedometer to read more accurately and will change your ride height so you may want to compensate by repositioning the front forks.
The 120/70 tire is approximately (not all 120/70s are alike) 24mm larger in diameter than a 120/60, so to keep the identical chassis geometry, you'll need to raise the forks (drop the triple clamps) enough to compensate for the ~12mm higher axle. Take a front ride height reference dimension first.
The decreased loads to the handlebars and increased sidewall flex combine to give you less (better?) feedback on the state of the front suspension and the traction condition at the front tire.
All this may better suit your riding style. Or not.
Some, like myself, find a less-nervous front end to be more appealing for the road so we run a 70-series tire. On the track, the better road surfaces make the benefits of a lower profile 60-series tire more obvious.
Finally, if you intend to convert to mag wheels, for the street a 120/60 is less able to handle rough pavement (less sidewall to cushion and flex for shock absorption) so wheel rim damage, especially to mag wheels, is an important consideration.