I would say a big fat NO to this... it is NOT true that specialised ship always are more efficient... that is only true in some situations. Putting beam weapon on all ships is a way to secure your ships for being singled out in a beam fight within a combined arms fleet. Sure, a fleet of 10 ship at a total weight of 200kt who all have beams as secondary weapons will loose big against a similar large fleet all armed with only beam weapons. But that is assuming the beam fleet can reach the combined arms fleet into beam combat without losses which is far from a given.
Specialised ships is good if the ships are small enough but as they grow bigger there really are very little reason to not include a bit of everything on a ship as you will likely need all those capabilities in your task-forces anyway.
If you rely solely on pure PD, AMM ships then single them out in a missile engagement can become catastrophic as for each escort destroyed the fleet looses allot of defensive power... if all ships in a task force have PD, AMM to some degree you have allot more hull and armour to degrade your defences.
I think I should clarify: I think having PD, AMM, both in addition to the main weapon are fine and do not make a ship "multi-role". As you say they are always necessary and a warship intended to operate alone simply must have these.
In my mind "multi-role" deals with the main weapon (including fighters) i.e. how is a ship expected to deal damage - a ship which mixes ASM, beam, fighters all together will not excel in any single capability which will render it exploitable by an opponent who does excel all things being roughly equal (admittedly, rarely if ever is this the case). There is a reason even the massive Nimitz class do not mount cruise missiles, and why the Iowas didn't have a flight deck and CSG apiece.
The same is true in a beam confrontation... If I have 30 lasers distributed on ten ship versus someone who has 40 lasers on three ships (assuming the other seven ships in that fleet are useless now) and the ships are otherwise roughly equal then the side with less lasers will still win as they have more armour and hull for each weapons destroyed.
I would argue that this is not a failure of specialization but rather fleet composition and/or tactical and operational execution. If one takes a fleet of ten ships and finds oneself in an action where seven of his ships are functionally useless (and not merely underperforming), something has gone wrong well outside of the actual ship design process. Though again, this is assuming that all vessels have at least rudimentary PD, thus dedicated PD ships are not the bulk of a fleet and even those present should be able to contribute - at least AMMs can provide screening fire although beam PD may not be able to close the range.
Notably an advantage of specialized ships is that fleet composition can be tailored for the mission theater easily. A fleet of multi-role ships requires that every fleet have multiple capabilities in proportions as-built, regardless of what is called for. Again if 70% of the fleet is useless in a battle this is a failure of scouting, mission planning, and/or tactics rather than ship design.
I also don't think 3000t hangar space is excessive... I think it is too small. A ship of this type need allot of scout crafts in their hangar and then you need strike crafts as well. This ship likely will want to stay out of harms way as much as possible as it's first line of defence.
I 100% agree on the need for scouting crafts, however the mix of strike crafts and beam primary armament is what I cannot get behind, and as Borealis did state a desire for fighter squadrons that is the paradigm I'm addressing. I would sooner have a class suited for fighters and another suited for beam brawling than a mixed class that excels at neither, and then mix and match those classes to achieve the fleet composition for the mission at hand. Same if we consider ASM vessels as well.