Has there been significant evidence or reason to believe that speed chess will make you a worse regular chess player overall? Or is it purely anecdotal?
Rather anecdotal for chess particularly. Maybe there will be some meaningful (meta-) studies in future since due to excessive upcoming online speed chess within the last years, the data fundament for such studies is there in general.
Besides that, your question itself might be branched and distinguished further on since there are potentially at least two relevant aspects here: Absence of specific classical training vs. adaption effects from speed-specific training, that might influence your general habits in terms of position analyzing. The latter one is also obviously heavily associated with discipline. Due to the fact that many great classical players are also very good in speed-chess, the potential aspect of adaption effects doesn't seem to be that important at least as long as the classical abilities are trained sufficiently too. So it's likely not a question about too much speed-chess but rather a question at least about the absolute value of sufficiency of classical training.
A well studied aspect of cognition is, that the human brain's ability in pattern recognition and storing details is extraordinary, even for distinguishing almost identical patterns (it's quite bad in reconstructing them from scratch although...). It's trivial to see, that a familiar position is almost always a better fundament for further analysis than a totally new one (stressors...) and with speed-chess, you're able to memorize thousands of positions within a short time range in general, at least subliminally. For too familiar positions, this needn't to be an advantage and could even switch into the negative since commonly known issues from overtrained neuronal networks might be influential here too. Further on, familiarity is only a fundament here. Classical deep analysis discipline is obviously required further on.
Assuming that the individual classical training is "sufficient", this aspect theoretically emphasizes, that speed-chess can be a productive additional training even for pure-classical players (better awareness for non-deep tactics deep analyzers might sometimes miss, see for instance Kramnik's famous blunder of the century...). Worth to mention further on, that time-drouble is an often seen issue in general, so a good speed-chess performance can be a further positive constraint at least for those situations, enhancing your general self-confidence in terms of time-management.