Mostly true in terms of margins, at least if you consider finance and insurance separately from vehicle sales (as all dealerships do).
Nevertheless, new vehicle sales including F&I is a non-negligible factor in most dealerships' bottom line, and new vehicle sales can also be a significant driver of service volume (both organically and as a result of loyalty incentives).
You're also correct in assuming COVID has been good for new vehicle sales profitability.
Here are the per-segment breakdowns reported by the three largest dealership groups in the US, before and after COVID:
1. AutoNation (AN)
2019 Rv: 52.3% New, 25.6% Used, 16.7% S&P, 4.8% F&I, 0.6% Other
2019 GP: 14.3% New, 10.4% Used, 46.1% S&P, 29.0% F&I
2021 Rv: 46.7% New, 33.4% Used, 14.3% S&P, 5.4% F&I, 0.2% Other
2021 GP: 24.3% New, 13.9% Used, 33.8% S&P, 28.0% F&I
2. Penske Automotive (PAG) (excluding non-retail auto dealership businesses)
2019 Rv: 45% New, 35% Used, 6% Fleet, 11% S&P, 3% F&I
2019 GP: 23% New, 12% Used, 1% Fleet, 43% S&P, 21% F&I
2021 Rv: 44% New, 38% Used, 5% Fleet, 10% S&P, 3% F&I
2021 GP: 27% New, 17% Used, 2% Fleet, 34% S&P, 20% F&I
3. Sonic Automotive (SAH)
2019 Rv: 47.0% New, 35.0% Used, 13.0% S&P, 5.0% F&I and Other
2019 GP: 15.0% New, 10.0% Used, 44.0% S&P, 31.0% F&I and Other
2021 Rv: 41.3% New, 39.3% Used, 11.3% S&P, 8.1% F&I and Other
2021 GP: 24.1% New, 6.9% Used, 35.2% S&P, 33.8% F&I and Other
(Rv = revenue as percentage of total, GP = gross profit as percentage of total, New, Used, Fleet = vehicle sales, S&P = Service and Parts, F&I = Finance and Insurance)
(source: SEC filings)
N.B.: Gross profit may be a misleading figure in this comparison, as it excludes major expenses (notably, salaries and commissions for non-administrative staff) directly attributable to particular business segments. Alas, I'm lazy, and these are the only per-segment percentages immediately available in all dealership 10-Ks.
With all that said, I imagine many, possibly most, dealerships would remain comfortably profitable even if net (non-F&I!) profit on new vehicle sales were zero.
Finally, note that fixed vehicle pricing is not exactly a new idea: GM famously introduced the Saturn marque with a similar tactic. And as with this older example, I assume Ford isn't planning to mandate fixed pricing on financing, insurance, trade-in allowances, third-party extended warranties, dealership-installed options, or anything else beyond the selling price of the vehicle itself and possibly certain minor fees related to the sale.