Yes, it would be significantly more stable than a phonetic system. Depending on how you measured it, phonetic systems would likely last around a generation at most, but phonemic systems might last for upwards of a century before they become conventionalized enough to no longer be a near-1-1 match for the 'Standard' dialect (again, depending on how you measured it).

It's worth noting that English's current orthography is already mostly phonemic though (that is, it's a combination of multiple different systems which are themselves mostly phonemic), and that the vast majority of the world's alphabetic or syllabic writing systems are phonemic, not phonetic.

This is straightforwardly demonstrable in English by the use of, for example, the letter "t" for a wide variety of different consonant sounds in the same dialects: unaspirated and aspirated [t] (stop vs. top), released vs. unreleased [t] (stop vs. bit), glottal stop (batman), flap (butter), glottalized [t] (caught), etc.

Phonetic writing systems are very rare, because native speakers of a language are practically always unaware of their language's sub-phonemic distinctions. Virtually no American English speakers, for example, are aware that they pronounce the "t" in top and stop differently.

So, replacing current English orthography wouldn't even be "switching to a phonemic system". English already uses a phonemic writing system - you'd just be switching to a different phonemic writing system (though, admittedly, one that has fewer subregularities/"inconsistencies" (for now)).