It depends on the type of experience that altercation was. If it was you being confronted about photographing their kid without a model release, that's one thing, but if it was a more general unease of a photographer being present at the event, that's different.

For the latter, make sure you don't need a media credential (you probably don't) and get one if you do.

For the former, if your kid is competing, the odds are good that you already are acquainted with their teammates' parents so you can just ask directly, especially if you intend to share your album with them afterward.

My experience is that, with littler kids, if a photographer is not in the parent sideline/area, parents may wonder who they are. With older (high school) kids, they expect some media coverage so that part isn't a big deal. What they do care about is being able to reshare your shots to their socials or use them for other personal reasons. Depending on the high school or club, you may or may not need a media credential. If not, it's usually up to the coach (for high school) to decide whether you're allowed on/next-to the field/track/court. It's helpful to build a rapport with the coaches. It's also helpful to be able to show that you're a legitimate business and not just some rando.

In my case, I do events for free and provide full-res post-processed albums via Google Photos. This is a labor of love because I know athletes and their families (not to mention yearbook staff!) appreciate it. Maxpreps, SBLive and others contract with local photogs to cover events, too, and those sites aggregate and host the albums... but downloads average ~$20/image. It's not hard for a decent local photographer to favorably compete against those freelancers. Then it's also easier for me to upsell on portraits and media days. Media Days for school teams I typically charge ~$35/kid. For club teams it's usually $50/kid. For that they get a guaranteed 3 poses each plus leftover time for fun poses. Unlike a lot of commercial photogs, I charge this flat rate per athlete instead of a booking fee + per-image download or print packages. My experience is that they really just want digitals most of the time anyway, and even if I net less I don't really care because this is just a labor of love where I can cover my expenses and earn some spending money (~$10k/yr is acceptable given the time I'm putting into it).

Make up an official-looking "Event Photographer" vest, hat, lanyard with "Photographer Pass" on it, etc?

People generally ignore or even help workers with a bright vest, carrying a ladder, etc. So I imagine you would get a lot less suspicious looks doing something like that versus looking like an Average Joe.

Maybe with a polo shirt and embroidered made-up photography company logo and name on it.

You need to be extremely careful about taking photos of children without explicit permission from the parents.