PE to family office...how to think about this offer?

I'm nearing the end of my Sr. Assoc. year at a MF PE firm (e.g., Bain Cap, Advent, TPG) with a prior MBB consulting background out of undergrad. I just received an offer for a midsized family office (AUM in the single-digit billions) that spends most of its time doing traditional PE investments in the same industry as my vertical (industrials/services, but also a smattering of other verticals I'd be interested in), and I'm trying to think through whether I should make the leap. Some additional detail is below:

Title/comp: The original offer was for Sr. Assoc., but I'm negotiating a title increase to VP. Cash comp is almost exactly the same as what I currently make (low $400s k) but wouldn't include any additional carry-like incentive or co-invest, and the FO told me to expect only cash comp for at least the short to medium term. Cash comp was higher than expected, but am I right to be a little disappointed in no equity upside? I'm curious to hear what a market-rate offer would be for this role.

Team structure: There are 4-8 IPs on the direct investing side, split between two main offices (West and East Coasts), and I'd be sitting with a Principal/MD-equivalent (but not the group head) on one coast. All are quite senior, and I'd be the most junior IP on the team, at least for the time being. Given I've spent the last few years doing associate-type modeling and business diligence work, I'm a bit worried that this role could be more of the same, at least until the group hires analysts/associates under me in the next year or two (and the FO has told me it plans to hire 1-2 more junior resources over the next 18 months).

Mandate: I don't want to get into specifics here, but the FO's mandate is much broader than what I'm used to--$25mm equity checks on VC investments through $500mm+ on traditional LBOs. It's also more flexible in that it will either be brought in as a co-investor or source deals and recruit other FOs or other GPs to come in alongside them.

Part of me thinks this is a compelling offer: I get to be a part of a growing team in a vertical I'm interested in, but not limited by; I might have the flexibility to do the style of investing I'm more drawn to (earlier vs. later stage), and the cash comp to expected WLB looks better than what I currently have. But another part of me feels like this is too similar to my current role: same/lower comp with no equity upside, potentially will feel like a junior role for at least another year or two, spending most of my time working with one person/lacking the social office environment, etc. So I'd love a reality check from you all--does this sound like a really good offer, and I just have unrealistic expectations about family offices? Or would I be reasonable to push for something more (or keep searching altogether)? Thanks so much for your help.