Should I replace fillable PDFs?

Normally, I recommend replacing fillable PDFs with regular web forms

I think overall, yes, that would be the ideal solution, but you noted some of the the issues with this, which includes how to handle all the existing content already made.

We run into issues constantly because web browsers love to open PDFs in browser instead of opening the files in the native Adobe Reader application.

By allowing the browser to handle the PDF file, you are leaving it to each users individual browser settings (e.g. open PDF in browser, or download). You have no control this way.

What is standard practice for this sort of thing?

You need to take control of the entire viewing experience, by adding a PDF viewer that supports PDF form filling, and load the PDF in your viewer. This way you know exactly what the user is doing, and control the entire user experience, and you can test your corpus of files for testing against this particular viewer since you host it.

You could try to force downloading of the PDF, so that it opens in the default application for viewing PDF files instead of the browser.

See this question: How do I force files to open in the browser instead of downloading (PDF)?

There is no standard practise. You'll have to re-think your process:

Why do your customers have to fill out PDFs in the first place?

Do you just need structured data from the customer? Who needs the PDF later, a government agency? Do you have to provide the customer with the same PDF? Can your company create web forms instead of PDFs? Can you auto-generate the PDFs yourself? Does the customer need to see the PDF exactly as is when filling it?

At our company, this PDF-filling is part of our core business domain, so we have an in-house solution. If it is not crucial to your business, you could buy this PDF-to-web form bridge from DocuSign, HelloWorks et al.

The quick-and-easy way is to force PDF download and demand that your customers use Adobe Reader.

Edit/PS: As PDFs are two-dimensional landscapes, they can be hard to navigate on mobile devices. Zoomed out, you're able to see everything but can read nothing, or vice versa when zoomed in. You have to consider how many users are on mobile devices. The more mobile devices among your customers, the better a web-optimized solution that features real forms. After filling the form, you could present the user with the automatically generated PDF.