It does remove a lot of entropy. The number of distinct bridge deals is 5 x 10^28 (source https://jlmartin.ku.edu/~jlmartin/bridge/basics.pdf ). log2 of that is under 96 bits, which does seem small by today's standards of cryptography.
As an aside modern bridge dealing machines/algorithms generate a 96 bit random number and do a one-to-one map to a particular deal (ie not doing lots of random calls and moving cards with a Fisher Yates shuffle, say).