The pros of sticking with eToro are:
1. No Swiss stamp duty. You only pay Swiss stamp duty when you use a Swiss broker. Depending which eToro broker you use, you may pay a samp duty in the country it's domiciled in.
2. Lower brokerage fees. If you use eToro UK or Europe, you don't pay a brokerage fee for long stock position, as long as you don't use leverage.
The cons of sticking with eToro are:
1. Withholding tax. If you use US eToro but live in Switzerland, your dividends are subject to US withholding tax. You can claim part of it back by submitting a form which you should get from eToro. I believe the situation is similar for eToro Europe/UK. Either way, you will either lose part of your dividends or, in the best case, put a lot of effort into reclaiming them. In Switzerland, private investors don't pay taxes on dividends. If you are a dividend investor, this is a consideration.
2. Less access to Swiss markets. Swissquote lets you access niche Swiss stocks, like those listed on the Bern exchange.
If your main activity is buying US stocks for the purpose of reselling them, stick with eToro for that - in my opinion.
Benefits of using a Swiss broker:
If you buy to hold in order to earn dividends, using a Swiss broker would, in my opinion, be a better long-term option because you won't lose money to foreign dividend withholding taxes and won't pay tax on dividends here (assuming you buy stocks maybe once a month so you qualify as a private investor). If you hold very long-term, Cornertrader may be a better option as they don't charge custodial fees (but charge higher brokerage fees than Swissquote).
If that's the case, just use the comparison to find out what makes sense for you:
https://www.moneyland.ch/en/online-trading-comparison
Possible added benefits for you as a Swiss resident are bank customer privacy, the Swiss bank depositor protection scheme, the option of detailed tax statements designed for Swiss tax purposes, Swiss consumer protection laws, access to more Swiss stocks, local customer support and educational events.