Vintage worth the time to explore?

If you can find an older Sophia EL34 A/B for six hundred dollars or so and invest in a checkup by a qualified tech you'll be in great shape. The good news is that the JJ EL34s or 6CA7s are fine replacements and are not ridiculously expensive.

no, you will not get better performance from vintage amps compared to modern...

I had the Marantz 2238, the Pioneer sx-450, and the HK 430 all in my room and one after the other the little modest Marantz nr1200 bested them all in sound quality even with my vintage Epi 100 speakers.  All of the vintages were also already professionally restored, so no it's not the caps.  All of the vintages were sold... I was surprised, but there was no denying it: the new nr1200 was superior in all sonic respects.  My head to head testing cured my vintage jones forever.  

Plus, the nr1200 has all the modern features as well as vintage features like tuner, and bass/treble/balance knobs right on the front... check the thing out: it may solve all your problems for not that much money, around $600.  It's excellent, overall. 

It should match very well indeed with your ELACs, for sure.

Playing louder has at least 2 affects. For one our hearing response varies in the bass and treble with loudness. Both roll off earlier at low levels. Flat is around 100 dB if I recall correctly and that's too dam loud and can cause ear damage. Fortunately most software is engineered to sound good at lower levels. But mastering varies and the correct level can vary from recording to recording.

Another problem can be dynamic linearity(my handle). This isn't the ability to play loud cleanly(although that's a part of it). It's the ability to change levels accurately from micro to medium to macro, to not compress level changes. Unfortunately many recordings compress level changes. And the affect of this compression is most at low levels and playing loudly ameliorates this affect. Many speakers are guilty of dynamic compression. For lower level listening lack of compression is essential to sort of sounding live. Picking dynamic speakers is essential.

As others have noted, all gear sounds better when played louder, within their performance envelope.  Many will scorn their use, but if your Marantz has a loudness button, give it a try.  If you're satisfied with the result, then don't worry for now.  You can revisit this choice when your children are older.

Also, if you have heard gear played at your desired volume range and with the sound you seek, please let us know the details.

My personal take - if I could only play my stereo quietly, I'd probably own Magneplanars.  If you've only heard boxes, you should definitely check out Maggies or electrostats.

If I’m reading your post correctly, you are driving 4 speakers with one integrated amp, correct? That’s a big ask for most integrated amps. If lower volume/increased  lower frequency response is desired, perhaps try a set of Marantz MA-500 amps. They are available used on the regular. They are vintage at a lower cost. If memory serves, they can be daisy chained. Though I’m not sure your integrated has pre outs. This would be a lower cost way to see if the lower volume/increased lower freq response is doable. Without breaking the bank.