A 65-series tyre isn’t the way to improve turn-in.
It’s generally true that a 65-series tyre is shorter, so it lowers the front end. However, even tyres having the same aspect ratios have different heights. For example, the Pirelli Dragon Evo front tyre is almost 30mm shorter than the same size-rated Michelin Pilot Race (H2). So you should check this dimension before you replace your current setup.
If you decrease the tyre height then you lower the front end, and consequently, you lower the center of gravity (CG) of the bike as well. This also increases the fork angle.
If you raise the rear end using the ride height adjuster, or go to a taller rear tyre, you raise the CG relative to it’s initial position. So, lowering the front ride height or raising the rear ride height are not equivalent adjustments.
In particular, the thought that lowering the front to improve turn-in response is incorrect.
Ducati Corse, in a 1996 memorandum*, recommended raising the front 10mm to increase “flickability" as in a chicane. Yes, I said RAISE, not lower. Raising the front end raises CG and a higher CG makes the bike go to the tyre edge quicker (according to the Ducati race engineers).
* Ths memo was previous available via the search engine at the old Ducati.com web site but lost when they put up the new site and no longer available.
The same advice is currently given in the factory race bike setup manual.
Take a look at the Mille SP. It has the capability to raise the engine in the frame to increase CG to improve flickability. Same effect. Even the Mille R has the engine higher in the frame to do the same.
It’s true that when you move from a 65-series to a 70-series tyre you’ll sense a slower steering. But, the slower steering you sense is the effect of the change in the tire section, not from the higher front ride height. The higher 70-profile tyre flexes more, reducing the severity of the loads transmitted to the wheel rim, suspension and handlebars. So the steering feels more vague with a higher profile tyre.
Your tyres are an integral part of your suspension, so when you go from a 65 to a less-stiff 70 profile, you’ll often see some benefit from increasing fork compression damping a little.
You can also increase rear ride height further to compensate.
Another effect from the taller tyre is an increased rake to the forks, so you can try compensating by using the steeper steering head position. This will also give you a 6mm shorter trail but will make the bike more prone to head-shake.
But 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 tyre), the front tyre 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 tyre reduces this wear-induced effect.
Further, a lower 60-ish profile tyre (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 tyre will be kinder to magnesium wheels.
Also, an increase in the outside diameter of the tyre 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 tyre 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.
That said, I personally like the 70-series tyre on a superbike.