Hi there, I'm Matt Sergei (call me Matt, please) - Leaflet have all required events programmed in it and usually a developer has to disable certain event to occur like dragging the map with one finger (when the user intention is to move down the page).
So I actually have mixed feelings watching your video and what you want to achieve. Assuming you are using drawing plugin(s) and your developer has scripted the measurement tool.
Either way, specify (or I will/can help) what Leaflet features do you want to use on a mobile phone and we can together implement that.
Otherwise I’m a webmaster and developer for 26 years, implementing web animations/charts/maps/visualizations with SVG, D3.js, Leaflet and GSAP (and other) libraries for 7+ years.
Regards and hope you're safe,
Matt