I just picked up a Princeton Tec FRED, not super bright, but has pretty long advertised runtimes off of 3 AAA batteries. I haven't used it very much at all yet, but I have been using a Princeton Tec Remix MPLS for the past 5 years or so several times a month. It just started acting up and I need to send it back to P-Tec to get looked at.
There's several different rechargeable models out now.
Used to sell a shitload of headlamps working at REI for either climbers, cavers or runners. Where I was had a heavy runner market. The rechargeables tended to sell well and not come back as they were running every day. Black Diamond and Petzl both make good headlamps, just depends on the features you want. I've used Petzl since the Tikka's first came out back 15 years ago, but BD is also excellent.
I have run for years with a Princeton Tec Apex, since replaced by the Princeton Tec Apex Pro. It is one of the heavier/bulkier options out there, but for me at least, the battery box balances things out. It user CR123 batteries, but they seem to last a LONG time, especially in LED mode. And if you got rechargeable CR123s, it's more economical.
I hate headlamps. Like to light up where I am walking, which points down so much it's shining past my face oddly, and my head moves a lot so it's jittery light, gives odd effects to me.
So I wear a Streamlight Clipmate USB, on shoulder straps. Usually have a tiny pack or Camelbak, so plenty of room to stick it. Direct the little flexi head as needed (when faster on roads, more forward, when in twisty ankle-breaker woods, more down) and the light is more stable coming from my body, less glare as it's all happening below the eyeline. This goes 24 hours at low power, but that's official; I have never run out of battery doing anything with it, ever. Charging mine before a thing this weekend for the first time in... years?
If that's too weird, our other family headlamp (we have several) is the UCO Air. Both these are USB rechargeable, btw. Super light, good headbands, great switchology, but does only last a couple hours. Never really bugged me as I can't run very far anymore but I do wish they made one just like it with more battery.