Those of us who understand the IRS procedure know that the return goes through the same levels of review, whether e-mailed or paper, and the only difference in processing time is delivering and opening the mail (for which they are now caught up). IRS doesn’t say that e-filed amended returns are processed any faster: “As of June 24, 2023, we had 1.35 million unprocessed Forms 1040-X. We are processing these returns in the order received and the current timeframe can be more than 20 weeks.”

https://www.irs.gov/newsroom/irs-operations-status-of-mission-critical-functions

Penalties and interest are assessed only to the date of payment, so the best way to minimize those is to pay when you file the return, either by mail or electronically.