The main reasons Harrison signed with the Pats were the chance to play for Belichick and for the organization's winning pedigree. The words Tom and Brady are conspicuous in their absence in that tweet.
Also, please show me any legit source that indicated that Baltimore offer him “way more”. I could be wrong but I’m pretty sure that you made that up.
“The Patriots aren’t attractive to free agents without Brady” has already been regurgitated on multiple occasions in multiple threads so far this off-season, just like last off-season. For many posters, it is just a slightly more subtle way to keep the Brady over Bill argument alive and also to keep Twommy as part of their lives 4EVA. It is also a bullsh*t argument, with the possible exception of free agent skill position players.
Just like is was for Harrison, BB’s reputation amongst players & the organization's winning pedigree were the main draws (other than money, of course) for many of the free agents. They don’t need Brady, they need to recapture that winning pedigree and if they have pay a little more for key free agents in the present than they would have a few years ago (in relative terms) in order to get back into the legit contender tier, so be it.