That lines up with my experience too. Personally I normally do the following:

If a repeat visit to a hotel where I have a relationship, I always book direct and talk to management in advance which has consistently paid off (though this only applies to 2-3 hotels for me).

If a last minute and/or 1 night booking, I'll often book through FHR or an online Virtuoso agency like Skylark. If it's just a quick booking one doesn't have to worry about your agent not being available or charging a fee for a last minute booking, and one can still get the basic amenities.

Up until I got the Citi Prestige card, for all other bookings I tended to use a TA since they're able to spend more time on customizing your stay and trying to arrange something special. Whether you'll get something extra on top of FHR benefits really depends on your agent and the hotel. I've received great upgrades through FHR at certain properties, and sometimes no upgrade (when there was availability) when using a TA, but more often than not with a good TA I've been able to get a nice upgrade (often confirmed in advance which FHR won't try to do) and other amenities.

I'd say give a TA a try (there are many good ones on this forum), it's very unlikely you'll get a worse experience than with FHR and more likely that you'll be pleasantly surprised