It’s a valid question.
A funny way to answer would be that it depends on what miniatures you use. Table 12-1 of the v3 Rules states that Infantry Class is “Any unit represented by a soldier (infantry, marine, militia, airborne, etc.)”. In our game, we use HBG’s Axis minor soldier with a Panzerfaust, so it’s definitely an Infantry unit!
However, the SS Panzer Grenadiers are clealry in the Specialist Infantry class, as you rightly point out, on the Germany National Reference Sheet (v.3.3), same as the other specialized infantry units for Japan (SNLF), FEC (Ghurka), USA (USMC) or the Commonwealth and Italy (Colonial inf.).
They are a elite infantry unit. The fact that they only become available with Advanced Mechanized Infantry seems to me like a game mechanic to delay their arrival on the battlefied, not a definition of their unit class. In reality, the Panzer Grenadiers units were organized in combined arms formation, so there is also a logic to the fact that they cannot be build before you have obtained that technology. They would really be overpowering otherwise.
In addition, if they are not infantry-class, I believe that they would have been listed under the Tech Advances table, just like the Katyusha Rocket Artillery for USSR that requires Advanced Artillery.
The +1 attack bonus that is given by Artillery pairing applies to all infantry-class units under Rule 12.2. I see no reason exclude the SS Panzer Grenadiers from this, just as it could benefit Mountain, Marine, Airborne Infantry, Colonial Infantry, Ghurkas, SNLF and USMC.
However, Motorized and Mechanized Intantry are in the Vehicle class, as per Rule 12.2. You question makes me realize that they should not benefit from being paired with artillery despite their name.
Finally, Naval Transports can carry either two infantry-class units or one infantry-class unit and one other land or air unit (who does that for air units?). It should transport two SS Panzer Grenadiers, same as two Marines.