However, If I buy a new computer/laptop, create a brand new 2-of-3 multi-sig wallet in Electrum, import the three seed phrases, and then sign a transaction from that single wallet, would it be ok to use that computer later on as my main computer (with Internet access)?
Presumably you would want to move all the coins out of this broken multi-sig wallet to a new wallet set up which you can access properly. Once you have done that, then there is no major security risk to that device then going online, since the multi-sig wallet will be empty.
There is still a risk to your privacy if you were to leak the addresses from that wallet (if that concerns you), or a risk to the coins if you were to accidentally receive some coins to this old multi-sig set up in the future. I would probably just format the hard drive once I was done and start fresh. Alternatively, boot the new computer to a live Linux OS for the purposes of restoring your wallet, and then boot to the hard drive once you are done and want to use it as a regular computer again.