Gotham city is the Wall Street of DCU, there are a lot of millionaires and billionaires there. Bruce Wayne isn't the only person in Gotham who could afford the Batman's arsenal, nor is he the most likely.
Even if Bruce was suspected to be the person who provide Batman his toys, people are NOT likely going to think Bruce Wayne is the Batman. They are too dissimilar personality wise, not to mention, the general consensus of the Gotham public is that the Batman is 6'4 while Bruce Wayne is only 6'2. He is too short to be the Batman, lol.
Personally I think the worst is when some point when DC decided somewhere during the 2000s that most people didn't even know Batman existed. How the hell does that even work when he spent over 20 years on the Justice League!?
That doesn't seem like the same idea to me. Aquaman is somebody judged by people who pick up on memes that wore themselves before the internet came into existence, Batman I was talking about a change that was made to how he was perceived in-universe that makes no sense given how long he was around for, especially on the Justice League.
Trust me no one suspects Bruce Wayne is Batman. That idea is as ridiculous to Gothamites as the idea of Ben Affleck playing Batman is to us. Here's a video that shows what must go through a Gothamites mind when it's suggested that Bruce Wayne is Batman:
He totally doesn't sleep all day; he's got a business to run--plus he has to be Batman during the day sometimes. He's shown, during Grant Morrisson's run on the Batman comic, to use micro-naps (if the art is to be believed they're about 5 minutes long o_O) and in Beware the Batman he claims to only need 4 hours of sleep and is shown buying bovine glands, or something like that, to help get it down to 2 hours. He's just awesome that way. I could totally see him "sleeping in" for 2-4 hours after dawn before going to the Wayne Enterprises office. Plus, that's a good point about everyone assuming he parties, lol .