I hate to be the bearer of bad news....
But I'm the bearer of bad news. I had a Heritage 410HLF for 3 years. Bought it new and had high hopes. I could never get a clear sound out of it, and you really hit it on the head with "It feels like the am pis almost unable to control the cab." That is exactly what I determined with mine.
I used a number of amps in my quest to get usable sounds out of it: Mesa TT800, GK Legacy 800, Aguilar Tone Hammer 500, Warwick Gnome iPro, TC BQ500, MarkBass BlackLine 250, Trace Elliott ELF.
It's that upper mid articulation that's completely missing on this cab... I'd love to see a graphic response chart on it. I'm imagining a massive dip around 800-1,000hz. Like... HUGE, and not able to be overcome by amp EQ settings. It's like the ability to make these frequencies is just MISSING. Even when paired with the GK Legacy (a notoriously "sharp" and mid-heavy amp) I couldn't get a clear tone out of the thing.
I initially loved the cab. It sounds FANTASTIC in a home-practice setting, where I used it for the first year. Thick, rich, deep, and a great kick in the gut when you wanted it. It's when I took it to play at church that the problems began. I used to play at a smaller church where I was the only amplification, as our 20+ year-old sound system (designed and installed before drums, electric guitars, and bass were part of the service) simply couldn't keep up with outputting low frequencies at any volume. Suddenly, that "bedroom tone", or in my case "basement tone" was useless. When trying to mesh with piano, guitar, drums, and 2-5 vocalists, all note clarity on the bass was gone. My EQ attempts to get some definition were futile. I found myself turning up the volume to try and hear what I was doing but then it all just became a low, woolly, overbearing, and ear-fatiguing mess.
Crazy enough, I was replacing an older (early 2000's) Ampeg B series (the cheap line) 410HLF with this new "improved" Heritage cab. The B series was the clearer cab! After a month of trying to make it work, I sold both Ampeg cabs and got the Mesa Subway 115 and 210 cabs. What a delight!
I will give this caveat... I never tried the Heritage cab in a rock band setting. I didn't start gigging with a classic rock band until a couple years after I sold the Heritage. Maybe in that setting with some overdrive in the tone, the cab shines like a diamond! I just never got to try it in that setting. Separately, in all of this I had a back blowout and major surgery. That ended my days of schlepping 80+lb cabinets! My new physical limitations were a major driver in the choice to cough up the cash for some premium lightweight cabs.
Best of luck in your search. I'd really love to hear that you find some great amp/cab combination that satisfies your bassic desires! I still have a soft spot for those big Ampeg 410s, even though they never quite worked for me.