So the reaction you are talking about is natural behavior of lawyers being exposed to legal documents and contracts. If there is anything that could be interpreted to impact the company, their job is to consider it and think "what-if".
The question comes down to, what is a healhty way to handle the result of lawyers paranoia. Best-practice is to do a cost-benefit analysis and balance the benefits with the insight of the legal advice. Second-worst is to avoid anything with a risk, regardless of benefits, in order to avoid it. Worst choice is to ignore the lawyers. Most companies, including the "very well known technology companies", pick the second-worst option for anything that is not critical to the company survival. Its clearly not the best option, but it keeps the status quo.
The fear that our lawyers have is that, since putting up the software in a service counts as a derived work, our whole software stack (including the stuff we don't open source) will have to be opened along with it. There have to be clear service boundaries between the AGPL software and the stuff we write ourselves, and the lawyers don't trust us to write in appropriate boundaries.
It's really kinda tragic, because we actually do submit source code upstream when we make changes to open source software that we run internally. As in, if it's an OSS product that we just use for some dumb internal automation thing, we'll submit patches if the license is BSD or MIT, but as soon as GPL (especially AGPL) hits anything suddenly the lawyers get paranoid because of what constitutes a "derived work", which can be interpreted as anything that links against the software to make a complete product.
The upshot of this is, if the OSS software is on an unrestrictive license like BSD or apache, we contribute upstream. If it's GPL or especially AGPL, we simply don't touch it, ever.
[1] A very, very well known technology company.