Not to argue on this, but pact of chain grants the warlock ability to immediately have access to these special familiars and as simply stated in MM these familiars grants Magic Resistance (within 10ft).
Ever wondered why such creatures in MM that have these familiar option are all in the Warlock's special familiar list? coincidence? i dont think so.
As there are no hard and fast rules regarding this, it's open to DM's interpretation. Note that not all of the four special familars available to the pact of the chain warlock get MM sidebars, just three (imp, pseudodragon and quasit). The wordings are similar (imp states that it "can enter a contract to serve another creature as a familiar", while pseudodragon and quasit entries state they "can serve another creature as a familiar"). Benefits are the same for all three (improved telepathic link (1 mile instead of regular familiar's 100 ft.), and magic resistance if within 10 ft. of the master).
Sprites don't get the sidebar. They also don't have the magic resistance. And, if you choose to use the MM sidebars for chain pact warlocks, don't get the improved telepathic link.
Find familiar spell gets you an animal familiar from a specified list, the familiar retains its animal Intelligence score (generally 1 or 2, IIRC); the familiar can act independently, but cannot attack; has a 100 ft. telepathic link with master; can be dismissed in a pocket dimension; can assume other forms by repeated castings of the spell; can deliver touch spell attacks; and generally have only a few hit points.
The
main benefit of the chain pact (IMO, of course) is to let you have a more intelligent, tougher (more hp) and generally more capable familiar. The fact that
all improved familiars are included in the PH and that the sidebar is not included anywhere in the PH clearly implies that the chain pact warlock is not meant to get MR resistance (or improved telepathy, but that's just gravy compared to the MR).
MM sidebar (again, IMO) is there to let the DM use it for NPCs, or as a special reward for a spellcaster PC (including clerics, druids, eldritch knights, arcane tricksters, etc.)
And this option is from DnD official monster manual. Its was not from a canon source. As a player I dont question a DM if his monsters and its traits are not in PHB. Players can have access to monster manual too and use it as their resource.
Again, I'm not saying your interpretation is wrong. However, the creatures in the Appendix D in the PH are there so the player's wouldn't have the need to consult the MM. You get animal companions, familiars, animal forms to wild shape into and skeletons and zombies that you can create with
animate dead. It looks like (although I'm not 100% certain) you get the majority of permanent companions (as opposed to stats of creatures you can summon via spells)
Besides this is just magic resistance, it doesnt make chain warlocks invincible or breaks the game. A smart DM will just hurt them thru physical attacks or kill the familiar.
While magic resistance in 5e is not as good as it was in older editions, it still represents a
very nice boon (advantage to saves vs. spells and other magical effects). The only other class that gets a similar ability is a 14th level abjurer, but they get only
spell resistance (advantage to saves vs. spells, not including "other magical effects").
Try to compare it to other pact boons. Blade pact gives you a melee weapon, which is considered magical, and you get proficiency in it. Tome pact gets you three cantrips. Both of those benefits are nice, but nowhere close to magic resistance. Chain pact basically gets you a slightly improved 1st-level spell (
find familiar). That (IMO) is exactly in line with other boons.
In short, magic resistance won't break the game, but it far outweighs other benefits gained at lower levels.
Regards.