"Switzerland and the EU have a close economic relationship based on a series of bilateral agreements, giving the country direct access to parts of the EU's internal market including the free movement of people"
Indeed they do. But what's missing from that quote is the fact the the EU is unhappy with that arrangement and has been pressing the Swiss to agree to an all-encompassing "Treaty". The reason for this is that under the current system, the Swiss can opt out of individual aspects of their relationship with the EU as it suits them (and they have on occasions in the past). The EU doesn't like that idea because they have no control over it. They want a Treaty that covers everything where the only opportunity for opt outs would be to withdraw from the Treaty entirely. This would put the Swiss in a difficult position because, as EU members have found, there are often aspects of membership that do not suit, but they are stuck with it or leave.
Talks between the EU and Switzerland on this issue have been going on for donkey's years but broke down about eighteen months ago:
https://www.bbc.co.uk/news/world-europe-57251681
The result of this is that the bilateral agreements will either expire (if they are time bound) or the EU will withdraw from them anyway because it essentially wants the Swiss to be subject to all of its legislation as if it was a member. Any UK government getting involved in anything similar needs it bumps felt.