It's almost correct, but not quite.

There are two separate, unrelated ways to get a familiar in D&D 5e, and you're trying to combine them.

The first method is by use of the Find Familiar spell, which allows you to summon a celestial, fey, or fiendish spirit that takes the form of any of a list of creatures. This list is expanded by the warlock pact of the chain. This familiar is perfectly obedient, can be resummoned when it dies, can be hid in a pocket dimension, deliver touch spells that you cast, and everything else specified in the spell description. This method gives you a familiar with the basic stats of the chosen creature, not the "variant: familiar" traits of the chosen creature (unless your DM chooses to have that creature type appear).

The second method is by finding a quasit, imp, or pseudodragon that has the "variant: familiar" trait (which is 100% up to the DM), and enlisting it as a familiar by interacting with it. This familiar has only the traits listed in the stat block for that creature, including the variant traits, but none of the traits of familiars given by the Find Familiar spell. No pocket dimension, no touch spells, no limitations on what actions it can perform, and if it dies, it's just dead. This type of familiar is an NPC controlled by the DM (much like a hireling or other follower), and is only as obedient to the PC as the DM says it is, using the MM entry as a guide.

Now that we've established how things actually work, we can address your real concern. Your warlock can't yet communicate at a great distance, but he can soon. There is a warlock invocation available to him called Voice of the Chain Master that does the same thing, but with unlimited range on the same plane. As you've realized, this ability has some incredible potential, especially for scouting.

It's not any more powerful than other options and should not be limited.

Let's compare it to some other invocations. There's one that lets a warlock cast Disguise Self as at will. This would let him see a guard, and appear exactly like that guard and just walk around the enemy camp unimpeded. Or maybe impersonate the leader of the camp and just take it over without even a struggle. There's another invocation that lets the warlock cast arcane eye at will, which gives you a way better scout than an easily killed creature. A familiar, even an invisible one, still has to succeed on a Dexterity (stealth) check to avoid being heard and then easily killed. An arcane eye does not.

So, in order to fully utilize this scouting ability, your warlock has to pick the chain pact and spend one of his few, precious invocations, both of which are huge opportunity costs. He deserves something in return. This something is you not limiting it. It already has a flaw in still being able to be killed by anything that hears it, or smells it. That's right, just about any pet wolf is going to be enough to catch this familiar. It doesn't need any more limitations.

There's nothing in the rules that prevents you from having the service of two creatures with the Familiar variant.

What you have to remember is that variant monsters, like monsters, are designed for the GM to use to make enemies more interesting. The Familiar variant is a monster variant, just like the troll's Loathsome Limbs variant or the Genie Powers variant. They're for GM use rather than player use.

The Mage NPC in Appendix B of the Monster Manual also has a Familiar variant, which says:

Any spellcaster that can cast the find familiar spell (such as an archmage or mage) is likely to have a familiar. The familiar can be one of the creatures described in the spell (see the Player’s Handbook) or some other Tiny monster, such as a crawling claw, imp, pseudodragon, or quasit.

So the Familiar variant is for GMs to create more interesting NPCs, rather than to provide players with additional options. Of course, with your GM's permission, you could obtain one of these familiars. This would probably involve actually finding such a creature and somehow forming a bond with it. But this relies solely on your GM to allow and arbitrate.

Needless to say, if even getting one familiar this way is entirely up to your GM, getting two is, even more so. There's nothing in the rules to prevent it, but you'll have to talk your GM into it if you want to have a quasit on each shoulder.

Now for combining Find Familiar with the Familiar variant: The interpretation that causes the least difficulty is that the variant Familiar isn't actually a familiar, it just "serves you as a familiar". In this case, there's no interaction between Find Familiar and the Familiar variant, and everything is fine.

However, if the variant Familiar is a familiar, well...things get weird. If you have your familiar from casting Find Familiar, and you then bond with one from the Familiar variant, there's no way to tell what happens. You "can't have more than one familiar at a time", so you've already put the game in a paradoxical state. Your original familiar might vanish, or your new one might die, or, well, anything, really.

If you have a familiar from the Familiar variant, and you cast Find Familiar, then by the rules (when using the interpretation that variant familiars still count as familiars), you get to change the form of your variant Familiar. You probably don't want to do this, since all the forms you could change it to are weaker than the one you've got, but there it is.

The fact that these rules break down completely when faced with each other is just more evidence - the Monster Manual is not meant as a player resource. Sorry, but it's not. Every spell or ability (like Wild Shape) that would require a player to look at the Monster Manual says "your DM has the stats" or something similar. The Familiar variant was never meant for players to see. It belongs to the GM.