Yeah, that functions a little differently again, as it only applies to a roll which does not have PB applied. I don't actually know exactly how JoaT functions on the sheet, though I'm sure someone could rip the workings out. It could be possible to hijack it to apply it for the Stone, but I don't know how.
The workaround I use is combined with Chat Menus. You need to be using an ability check menu for your rolls for this to work, not straight from the sheet.
You have the ability macros in your Chat Menu linked to the default sheet actions: [Strength](~selected|strength). This will roll the sheet's inbuilt Strength ability check, unless you overwrite it with your own Ability called strength: (or you can reference on stored on a Macro sheet)
@{selected|wtype}&{template:simple} {{rname=^{strength-u}}} {{mod=@{selected|strength_mod}@{selected|jack_bonus}}} {{r1=[[@{selected|d20}+@{selected|strength_mod}@{selected|jack_attr}[STR]]]}} @{selected|rtype}+@{selected|strength_mod}@{selected|jack_attr}[STR]]]}} {{global=@{selected|global_skill_mod}}} @{selected|charname_output}
This is the standard strength ability check roll, just with the Skills global modifier field added in. You will now get any global mods to skill checks added to straight-up strength ability checks. Technically there are no "Skill checks" in 5E, just Ability Checks which can have you proficiency bonus added on via skill proficiency, so anything that is globally adding to a skill check is technically adding to an Ability Check. I'm not sure why the sheet has this upside down... there shouldn't even be a global skill modifier field at all, just a global ability check field.
Anyway, doing the above for the 6 main stats will give you Global Mods for your Ability Checks.
The bonus from Stone of GL still needs to be added to Initiative, either directly or by adding the Global Mod field to the Initiative macro.