You have fingerprints, and you leave them on every surface you touch, in every place you go, and they stay behind so your previous presence can be determined.
Is that intrusive to your privacy? If so you can wear gloves.
A microchip will be hard to read from far (it won't be powered) so you have to be within several feet of the person, and you have to position yourself so they aren't between you and the chip (ie, the reader will make it through a little skin, but not your whole body), not only that but to read from more than a few inches the reader will be necessarily large (briefcase size or bigger), and power hungry.
But they can't detect where you've been after you've left, and the chip won't release confidential information (if designed properly). And if it really bothers you, you could wear gloves that block the RF, though if that's really a problem you might as well not have it at all.
As far as transaction privacy, all my phone communications, current location, browsing history, and credit/debit purchases are already tracked by the government. I have Alexa in my home, Siri in my pocket, and I'm on the internet 80% of my waking hours.
And you're worried about an RFID tag?
Part of the reason I may be comfortable with them is that I've designed systems to use RFID and I know how it works, its limitations, how secure it is, etc, etc. It's no more invasive that any other of the many personally identifiable artifacts you carry.
Shoot, my driver's license (and soon everyone's driver's license) has an RFID chip in it - the state helpfully provides an RF blocking sleeve, but your credit cards may have it, and your phone, if it supports bluetooth, wifi, cellular, or a non contact payment system, is way more trackable than an embedded rfid chip. I can buy a device that will identify all the cell phones within a few hundred feet, and tracking that over time I could associate the phones with various people and their phone numbers.
Unless you don't carry a cellphpone, driver's license, RFID enabled credit cards, and you mask your identity (face detection software is being used on most urban city cameras now, particularly around points of terrorist interest such as stadiums), and wear gloves, then an embedded RFID chip doesn't expose you any more than you're already exposed.