Reesa,
I would not assume that. The provision of the rule makes no distinction between servicers or anyone else, so it is important that the mortgage be assigned into the "right name" before starting the foreclosure.
Here is the language of the doc stamp rule:
"(25) Assignment of Successful Bid – An interest in realty transferred or conveyed by assignment of successful bid at a foreclosure
sale is taxable under Section 201.02, F.S"
There are also other benefits that these servicers may be losing for their investors by not doing assignments first. For example under Fla. Stat. s. 718.116, the 1% cap on condo assessments may be lost if a bid is assigned.