I doubt that a new CD player will sound less bright than your Rega. CD players have a flat frequency response and from that standpoint they sound very similar to one another.
I think what you are hearing is a rolled-off high frequency response from your new Ortofon cartridge. It is making your CD player sound bright by comparison. You don't say which presentation you prefer but if you like the less bright sound of your turntable then getting an equalizer to put between your CD player and your preamp may be a better answer.
FWIW, I'm running a HANA SL cartridge and it sounds strikingly similar to my digital rigs. The freqency response of the HANA is very flat. I have a couple DACs, a couple of transports, a CD player, and an SACD player and they have slight differences but none of them sounds brighter than the other.
It would be intersting to try the HANA and see if that sounds more like your CD player. Depending on the rest of your system the diminished high frequencies you are hearing from your ML/Ortofon combo may be preferable in which case you might want to reevaluate your speakers.