I have all of the beltfeds you mentioned, a 1919A4, a Shrike with an M16, an M60 as well as a HK21E/23E, MM21EK/23K, and an FM9. I also have the FNC as well.
If forced to pick between the 1919 and Shrike+M16 I would keep the Shrike+M16.
Sure the 1919 is more of a "real" machinegun but that also comes with a cost. The gun is big and heavy, generally requires a tripod and T&E, in standard form shoots a round (30.06 or 308) that is twice as expensive as 5.56. In my experience this all translates into the 1919 getting shot less frequently as it a lot more of a commitment to take somewhere and shoot vs. the Shrike.
The M16 (outside of the Shrike) is one of the top three most versatile machineguns you can own. So when you don't want a 5.56 beltfed it can be a standard M16, a short barrel M4, a 22lr machinegun, a 9mm submachinegun, a 9mm beltfed, a AM180 drum machinegun, etc. In contrast a 1919 is always going to be a 1919.
Where the 1919 shines for me is when you are taking it to a big machinegun shoot and you want a big heavy workhorse to fire thousands of rounds every day with minimal fuss. In that situation the trouble to haul a 30lbs machinegun in a 20lbs pelican, plus 30lbs worth of tripod and support gear/parts, plus 50+lbs worth of ammo etc. makes it worth it for an all day event to ultimately take a 150+lbs of stuff to the range for one gun. I virtually never take my 1919 out for a quick afternoon trip to the range and I even have mine set up in 7.62x39 so I can shoot cheaper ammo all day at roughly half the cost of 30.06 or 308.
The Shrike is much more dynamic shooting as it not fixed on a tripod and is pretty much to me like a belt-fed M4. It fits in a standard soft rifle case, doesn't need a tripod, T&E, weights less than 10lbs, etc. However, a Shrike isn't going to last as long from a round count perspective as I don't see a Shrike upper receiver going 100K+ rounds like a 1919A4 is able to do. Shrike parts are also a lot more expensive than 1919 parts/barrels so if this is going to be a high volume shooter than maybe the 1919 makes more sense from a lifespan and spare parts/barrel perspective.
The Shrike and even more so the M16 lower is just so much more practical shooter than the 1919 in most situations. I personally also prefer shoulder fired or bipod guns vs. sitting behind fixed guns mounted on a tripod. However again that is just personal preference.
Ultimately for you its going to probably come down to your use case and preference.
If you are more into big all day or all weekend shoots, the 1919 is probably a better option, especially when paired with a x39 kit to make it more on par with the Shrike from an ammo cost perspective. If you are more of a quick afternoon range trip person the 1919 probably isn't the best choice and is going to quickly wear on you dragging it and all its associated equipment to the range for a couple hour session.
Hope this helps.
My personal 1919 at last springs Big Sandy shoot.