what is my system's weakest link?

A decent tube pre-amp might be the ticket to a somewhat warmer sound with the Thiels. A juicier amp to go along with it to better address the rest of the goals perhaps.

Dynaudio would be one brand to consider for perhaps a tad warmer sounding option to the Thiels.

Adding a powered sub or two to the Thiels might help also, espcially if you think they may be undersized to deliver good extended bass in what sounds like a decent sized room.

Lots of ways to skin these cats. WIres might help fine tune things as well later if needed, but may not be the place to start in this case.

Moreytalmor, my suggestion to you is to go slowly. You seem to have a
pretty good idea of the type sound you are looking for. This is all a learning
process and it is most educational to change out one component at a time.
You seem to want to keep the Thiels; great, that simplifies matters. While I
think you are a prime candidate for tube electronics, I would get a better
turntable/cartridge next and see how far that takes you to your goal. A
better turntable will give you more of the musical involvement that you are
craving. In fact, I might start by upgrading the cartridge first to one that is
really good and you can install on the better turntable that you will buy later
after you get a handle on your new sound. Don't sell the old cartridge; you
will probably want to sell the Project with the old cartridge. Don't be afraid
to spend what may seem to be a disproportionate amount of money on a
cartridge; rmember any information that you lose at that stage of things you
will never be able to get back no matter how good the gear upstream. Start
by reading this:

eanlg&1200430667&&&/Who-needs-a-MM-cartridge-type-when-we-
ha>http://forum.audiogon.com/cgi-bin/fr.pl?eanlg&1200430667&&&/Who-
needs-a-MM-cartridge-type-when-we-ha

After all the advice here, the only way to figure this out is to listen to some different pieces. As Frogman says above, take it slow and go one piece at a time.
Your speakers are approaching 20 years old so I would still suggest you start there. There was a pair of Quad 12L2s listed on Audiogon today for $500. I have owned both the Thiel 1.5s and the Quads and believe the Quads have a much more exciting sound.
Although it can be a grueling experience, the best way to know is to experiment.

Dont have any!!

With a limited budget the Thiel 1.5 must go due to demands they make with amps and offer little in return. As mentioned by others look into Focal, Triangle or even a current Rega speaker RS3/5 and keep the Brio. If you must have a Thiel speaker look into a 2.3 or 2.4 for a long term system/project. IMO the 2.3/4 are the "Thiel" sweet spot but you will need patience and money to get them right.