I guess I've been incredibly fortunate with primers since I began handloading in 1965. With rifle or handgun cartridges, I try seat the primer fully which almost always means the primer is slightly below flush. It takes but a fraction of a second to run an index finger over the primer to check this; a little less convenient chore if using a progressive machine but it can still be done.
If using a handheld priming device make sure it's not worn out. A worn tool often prevents primers from being fully seated regardless of the amount of effort used.
With regard to revolvers in particular, I've yet to see an out-of-the-box unmolested revolver that would not fire any and every primer with 100% reliability. Change springs and get action jobs if you wish but you always take a chance on forfeiting reliability only for a somewhat lighter and maybe smoother trigger. Instead, consider practicing more with the as-is gun and you'll likely get used to it and be quite happy with it.
If a modified revolver requires a specific primer to work reliably, that's not really a reliable gun in the strict sense, but that's okay if that's what you want.